Riemann Sums

Approximating Area Under a Curve

Graphically, we can consider a definite integral, such as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\displaystyle \int_{0}^{1}x^{2}\, dx} }

to be the area "under the curve", which might be better said as the area that lies between the lines Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=0}   and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=1, }   the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x } -axis Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (y=0) }   and the curve Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y=x^{2}. }   In order to find this area, we can begin with a familiar geometric object: the rectangle. In this case, we wish to find an area above the interval from Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0} to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1} . In order to approximate this, we can divide the interval into, say Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 4} shorter intervals of equal length. Let's call this length Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta x } , and since they are all the same length, we know that the length of each will be Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1/4. }

However, in order to define an area, our rectangles require a height as well as a width. Most often, calculus teachers will use the function's value at the left or right endpoint for the height of each rectangle, although we could also choose the minimum or maximum value of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) } on each interval, or perhaps the value at the midpoint of each interval.

Let's approximate this area first using left endpoints. Notice that our leftmost interval is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [0,1/4], } so the height at the left endpoint is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(0)=0. } This means the area of our leftmost rectangle is

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(0)\cdot\Delta x\,=\,0\cdot\left({\displaystyle \frac{1}{4}}\right)\,=\,0. }

Continuing, the adjacent interval is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [1/4,1/2]. } Now, our left endpoint is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1/4 } , and our area is

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\displaystyle f\left(\frac{1}{4}\right)\cdot\Delta x\,=\,\frac{1}{16}\cdot\frac{1}{4}\,=\,\frac{1}{64}}. }

The next interval to the right is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [1/2,3/4], } and as such the left endpoint is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1/2, } so the area is

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\displaystyle f\left(\frac{1}{2}\right)\cdot\Delta x\,=\,\frac{1}{4}\cdot\frac{1}{4}\,=\,\frac{1}{16}.} }

Finally, we have the rightmost rectangle, whose base is the interval Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [3/4,1]. } This has Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3/4 } as its left endpoint, so its area is

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\displaystyle f\left(\frac{3}{4}\right)\cdot\Delta x\,=\,\frac{9}{16}\cdot\frac{1}{4}\,=\,\frac{9}{64}.} }

Adding these four rectangles up with sigma Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\Sigma) } notation, we can approximate the area under the curve as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{rcl} S & = & {\displaystyle \sum_{i=1}^{4}f\left(x_{i}\right)\cdot\Delta x}\\ \\ & = & {\displaystyle 0+\frac{1}{64}+\frac{1}{16}+\frac{9}{64}}\\ \\ & = & {\displaystyle \frac{14}{64}}\\ \\ & = & {\displaystyle \frac{7}{32}.} \end{array} }

Of course, we could also use right endpoints. In this case, we would use the endpoints Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1/4,\,1/2,\,3/4 } and 1 for each interval from left to right to find

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{rcl} S & = & {\displaystyle \sum_{i=1}^{4}f\left(x_{i}\right)\cdot\Delta x}\\ \\ & = & {\displaystyle f\left(\frac{1}{4}\right)\cdot\Delta x+{\displaystyle f\left(\frac{1}{2}\right)\cdot\Delta x+}{\displaystyle f\left(\frac{3}{4}\right)\cdot\Delta x+}{\displaystyle f\left(1\right)\cdot\Delta x}}\\ \\ & = & {\displaystyle \frac{1}{16}\cdot\frac{1}{4}+\frac{1}{4}\cdot\frac{1}{4}+\frac{9}{16}\cdot\frac{1}{4}+1\cdot\frac{1}{4}}\\ \\ & = & {\displaystyle \frac{1}{64}+\frac{1}{16}+\frac{9}{64}+\frac{1}{4}}\\ \\ & = & {\displaystyle \frac{15}{32}}. \end{array} }

Note that in this case, one is an overestimate and one is an underestimate.

Additional Examples with Fixed Numbers of Rectangles

Example 1. Approximate the area under the curve of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)=x^{3}-x } from -1 to 3 using Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n=4 } rectangles and left endpoints.

Solution. Note that our Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x } -values range from Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -1 } to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3 } , so our length is actually Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3-(-1)=4. } Thus each rectangle will have a base of

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta x\,=\,{\displaystyle \frac{b-a}{n}\,=\,\frac{3-(-1)}{4}\,=\,\frac{4}{4}\,=\,1.} }

This is our first step. This means our intervals from left to right are Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [-1,0],\,[0,1],\,[1,2] } and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [2,3]. } Choosing left endpoints, we have

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{rcl} S & = & {\displaystyle \sum_{i=1}^{4}f(x_{i})\cdot\Delta x}\\ \\ & = & f(-1)\cdot1+f(0)\cdot1+f(1)\cdot1+f(2)\cdot1\\ \\ & = & -2+0+0+6\\ \\ & = & 4. \end{array} }

Here is where the idea of ``area under the curve becomes clearer. We actually have a signed area, where area below the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x } -axis is negative, while area above the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x } -axis is positive.

Example 2. Approximate the area under the curve of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)=x^{3}-x } from Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -4 } to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 4 } using Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n=4 } rectangles and midpoints.

Solution. Here, our Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x } -values range from Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -4 } to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 4, } so

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta x\,=\,{\displaystyle \frac{b-a}{n}\,=\,\frac{4-(-4)}{4}\,=\,\frac{8}{4}\,=\,2.} }

As a result, our intervals from left to right are Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [-4,-2],\,[-2,0],\,[0,2] } and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [2,4]. } More importantly, our midpoints occur at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -3,\,-1,\,1 } and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3 } respectively; this is where we will evaluate the height of each rectangle. Thus

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{rcl} S & = & {\displaystyle \sum_{i=1}^{4}f(x_{i})\cdot\Delta x}\\ \\ & = & f(-3)\cdot2+f(-1)\cdot2+f(1)\cdot2+f(3)\cdot2\\ \\ & = & (-24)\cdot2+0\cdot2+0\cdot2+24\cdot2\\ \\ & = & 0. \end{array} }

Here is where the idea of ``area under the curve becomes clearer. We actually have a signed area, where area below the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x } -axis is negative, while area above the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x } -axis is positive.

Defining the Integral as a Limit

Although associating the area under the curve with four rectangles gives us a really rough approximation, there's no reason we can't continue to divide (partition) the interval into smaller pieces, and then get closer to the actual area. The graphic on the right shows precisely the idea. We can keep making the base of each rectangle, or Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta x, } smaller and smaller, and we'll get a better approximation. More importantly, we can continue this idea as a limit, leading to the following definition.

Definition. We define the Definite Integral of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) } on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [a,b], } written

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\displaystyle \int_{a}^{b}f(x)\, dx}, }

to be

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\displaystyle \int_{a}^{b}f(x)\, dx\,=\,\lim_{n\rightarrow\infty}\sum_{i=1}^{n}f(x_{i})\cdot\Delta x_{i}.} }

For an introductory course, we usually have Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta x_{i}=\Delta x={\displaystyle \frac{b-a}{n},} } so each rectangle has exactly the same base.

Using the Definition to Evaluate a Definite Integral

Frequently, students will be asked questions such as: Using the definition of the definite integral, find the area under the curve of the function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)=x^{2} } on the interval Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [0,3] } using right endpoints.

Rather than using ``easier rules, such as the power rule and the Fundamental Theorem of Calculus, this requires us to use the the definition just listed. The first step is to set up our sum. We have Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a=0,\, b=3 } and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n } is just an arbitrary natural (or counting) number. This tells us that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta x\,=\,{\displaystyle \frac{b-a}{n}\,=\,\frac{3-0}{n}\,=\,\frac{3}{n}}. }

For a given Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n } , our leftmost interval would start at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0, } and be of length Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta x=3/n. } This describes the interval Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [0,3/n]. } On the other hand, our next interval would start where the leftmost stopped, or Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3/n } , and it's length would also be Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta x=3/n. } This is the interval

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\displaystyle \left[\frac{3}{n},\frac{3}{n}+\frac{3}{n}\right]\,=\,\left[1\cdot\frac{3}{n},2\cdot\frac{3}{n}\right].} }

If we call the leftmost interval Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I_{1}, } then we would have Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I_{1}={\displaystyle \left[0\cdot\frac{3}{n},1\cdot\frac{3}{n}\right]}. } Similarly, our second interval would be Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I_{2}={\displaystyle \left[1\cdot\frac{3}{n},2\cdot\frac{3}{n}\right]}. } If we were to continue this rule, we would have that for any Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i=1,2,\ldots,n, } we could write Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I_{i}={\displaystyle \left[(i-1)\cdot\frac{3}{n},i\cdot\frac{3}{n}\right]}. } This allows us to determine where to choose our height for each interval. Since we are asked to use right endpoints, we would want Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f\left({\displaystyle 1\cdot\frac{3}{n}}\right) } for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I_{1}, } Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f\left({\displaystyle 2\cdot\frac{3}{n}}\right) } for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I_{2}, } and finally Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f\left({\displaystyle i\cdot\frac{3}{n}}\right) } for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I_{i}. }

This allows us to build the sum. For an arbitrary Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n, } we would have

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\displaystyle \sum_{i=1}^{n}f(x_{i})\cdot\Delta x_{i}\,=\,\sum_{i=1}^{n}f\left(\frac{3i}{n}\right)\cdot\Delta x\,=\,\sum_{i=1}^{n}\frac{9i^{2}}{n^{2}}\cdot\frac{3}{n}\,=\,\sum_{i=1}^{n}\frac{27i^{2}}{n^{3}}.} }

Using this result, we now have

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\displaystyle \int_{a}^{b}f(x)\, dx\,=\,\lim_{n\rightarrow\infty}\sum_{i=1}^{n}f(x_{i})\cdot\Delta x_{i}\,=\,\lim_{n\rightarrow\infty}\sum_{i=1}^{n}\frac{27i^{2}}{n^{3}}.} }

Now, we have several important sums explained on another page. These are the sum of the first Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n } numbers, the sum of the first Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n } squares, and the sum of the first Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n } cubes:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\displaystyle \sum_{i=1}^{n}i\,=\,\frac{n(n+1)}{2};\qquad\sum_{i=1}^{n}i^{2}\,=\,\frac{n(n+1)(2n+1)}{6};\qquad\sum_{i=1}^{n}i^{3}\,=\,\frac{n^{2}(n+1)^{2}}{4}.} }

Moreover, we have some basic rules for summation. These rules that make sense in simpler notation, such as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ca_{1}+ca_{2}=c(a_{1}+a_{2}) } or Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (a_{1}+b_{1})+(a_{2}+b_{2})=(a_{1}+a_{2})+(b_{1}+b_{2}) }

work the same way in Sigma notation, meaning

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\displaystyle \sum_{i=1}^{n}ca_{i}\,=\, ca_{1}+ca_{2}+\cdots+ca_{n}\,=\, c(a_{1}+\cdots+a_{n})\,=\, c\sum_{i=1}^{n}a_{i},}\qquad\qquad(\dagger) }

and

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\displaystyle \sum_{i=1}^{n}(a_{i}+b_{i})\,=\, a_{1}+b_{1}+a_{2}+b_{2}\cdots a_{n}+b_{n}\,=\, a_{1}+a_{2}+\cdots+a_{n}+b_{1}+b_{2}+\cdots+b_{n}\,=\,\sum_{i=1}^{n}a_{i}+\sum_{i=1}^{n}b_{i}.\qquad\qquad(\dagger\dagger)} }

Before worrying about the limit as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n\rightarrow\infty } , when we write Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\displaystyle \sum_{i=1}^{n}\frac{27i^{2}}{n^{3}},} } both Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 27 } and the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n^{3} } in the denominator are just constants, like the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c } in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\dagger). } As a result we have

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\displaystyle \sum_{i=1}^{n}\frac{27i^{2}}{n^{3}}\,=\,\frac{27}{n^{3}}\sum_{i=1}^{n}i^{2}\,=\,\frac{27}{n^{3}}\cdot\frac{n(n+1)(2n+1)}{6}\,=\,\frac{9n(n+1)(2n+1)}{2n^{3}},} }

where we applied the rule for the first Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n } squares. Finally, we can look at this as being approximately Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 18n^{3}/2n^{3} } for large Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n, } so the limit as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n\rightarrow\infty } is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 9. } Thus

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\displaystyle \int_{0}^{3}x^{2}\, dx=9.} }

A Left Point of View

What would change if we approached the above integral through left endpoints, instead of right? We would only be changing our value for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x_{i}). } For example, the leftmost interval is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\displaystyle \left[0\cdot\frac{3}{n},1\cdot\frac{3}{n}\right],} } so our left endpoint is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0. } On the other hand, our second interval is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\displaystyle \left[1\cdot\frac{3}{n},2\cdot\frac{3}{n}\right],} } so our left endpoint is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3/n. } For the interval Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I_{1}={\displaystyle \left[(i-1)\cdot\frac{3}{n},i\cdot\frac{3}{n}\right],} } our left endpoint is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3(i-1)/n. } Let's apply the same process to this value. We have

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{rcl} {\displaystyle \int_{0}^{3}x^{2}\, dx} & = & {\displaystyle \lim_{n\rightarrow\infty}\,\sum_{i=1}^{n}f(x_{i})\cdot\Delta x_{i}}\\ \\ & = & {\displaystyle \lim_{n\rightarrow\infty}\,\sum_{i=1}^{n}\left(\frac{3(i-1)}{n}\right)^{2}\cdot\frac{3}{n}}\\ \\ & = & {\displaystyle \lim_{n\rightarrow\infty}\,\sum_{i=1}^{n}\frac{9(i-1)}{n^{2}}^{2}\cdot\frac{3}{n}}\\ \\ & = & {\displaystyle \lim_{n\rightarrow\infty}\,\sum_{i=1}^{n}\frac{27(i-1)}{n^{3}}^{2}}\\ \\ & = & {\displaystyle \lim_{n\rightarrow\infty}\,\frac{27}{n^{3}}\sum_{i=1}^{n}(i^{2}-2i+1).} \end{array} }

From here, we use the special sums again. This means that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{rcl} {\displaystyle \lim_{n\rightarrow\infty}\,\frac{27}{n^{3}}\sum_{i=1}^{n}(i^{2}-2i+1)} & = & {\displaystyle \lim_{n\rightarrow\infty}\,\frac{27}{n^{3}}\left(\frac{n(n+1)(2n+1)}{6}+\frac{n(n+1)}{2}+n\right)}\\ \\ & = & {\displaystyle \lim_{n\rightarrow\infty}\,\left(\frac{9n(n+1)(2n+1)}{2n^{3}}+\frac{27n(n+1)}{2n^{3}}+\frac{27}{n^{2}}\right)}\\ \\ & = & {\displaystyle \lim_{n\rightarrow\infty}\,\left(\frac{18n^{3}}{2n^{3}}+\frac{27n^{2}}{2n^{3}}+\frac{27}{n^{2}}\right)\qquad\qquad(\textrm{for large }n)}\\ \\ & = & 9+0+0\\ \\ & = & 9. \end{array} }

Thus our choice of endpoints makes no difference in the resulting value.

A More Advanced Example

For most Riemann Sum problems in an integral calculus class, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta x } will always be the same width, and we will need to use the special sums to evaluate the limit. However, what can we do if we wish to determine the value of

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_{0}^{1}\sqrt{x}\, dx? }

Using rectangles of the same width as shown in the earlier animation would result in a very messy sum which contains a lot of square roots! This makes finding the limit nearly impossible. Instead, we could consider the inverse function to the square root, which is squaring. Instead of choosing Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta x=(b-a)/n=(1-0)/n, } let's consider Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_{0}=0, } and let

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_{i}\,=\,{\displaystyle \frac{i^{2}}{n^{2}}.} }

For example, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_{1}=1/n^{2} } while Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_{2}=4/n^{2}. } In particular, since we indexed the leftmost point as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_{0}=0, } this means that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\displaystyle \Delta x_{i}\,=\, x_{i}-x_{i-1}\,=\,\frac{i^{2}}{n^{2}}-\frac{(i-1)^{2}}{n^{2}}\,=\,\frac{2i-1}{n^{2}}.} }

Each Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta x_{i} } will be a different width, but either endpoint would be a square, so taking Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x_{i}) } will not leave a square root in our sum.

Now we have all the pieces. Let's use right endpoints for the height of each rectangle, so Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x_{i})={\displaystyle \sqrt{\frac{i^{2}}{n^{2}}}=\frac{i}{n}.} } Then

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{rcl} {\displaystyle \int_{0}^{1}\sqrt{x}\, dx} & = & {\displaystyle \lim_{n\rightarrow\infty}\,\sum_{i=1}^{n}f(x_{i})\cdot\Delta x_{i}}\\ \\ & = & {\displaystyle \lim_{n\rightarrow\infty}\sum_{i=1}^{n}\frac{i}{n}\cdot\frac{2i-1}{n^{2}}}\\ \\ & = & {\displaystyle \lim_{n\rightarrow\infty}\sum_{i=1}^{n}\frac{2i^{2}-i}{n^{3}}}\\ \\ & = & {\displaystyle \lim_{n\rightarrow\infty}\frac{1}{n^{3}}\left(2\cdot\frac{n(n+1)(2n+1)}{6}-\frac{n(n+1)}{2}\right)}\\ \\ & = & {\displaystyle \lim_{n\rightarrow\infty}\left(\frac{2n(n+1)(2n+1)}{6n^{3}}-\frac{n(n+1)}{2n^{3}}\right)}\\ \\ & = & {\displaystyle \lim_{n\rightarrow\infty}\,\left(\frac{4n^{3}}{6n^{3}}+\frac{n^{2}}{2n^{3}}\right)\qquad\qquad(\textrm{for large }n)}\\ \\ & = & {\displaystyle \frac{2}{3}+0}\\ \\ & = & {\displaystyle \frac{2}{3}.} \end{array} }

You will NEVER see something like this in a first year calculus class, but it is just a reminder that the definition includes the indexed Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta x_{i} } for a reason!