009B Sample Midterm 2, Problem 1

Consider the region Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle S} bounded by $x=1,x=5,y={\frac {1}{x^{2}}}$ and the $x$-axis.

a) Use four rectangles and a Riemann sum to approximate the area of the region Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle S} . Sketch the region Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle S} and the rectangles and indicate whether your rectangles overestimate or underestimate the area of Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle S} .

b) Find an expression for the area of the region Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle S} as a limit. Do not evaluate the limit.

Foundations:
See the page on Riemann Sums.

Solution:

(a)

Step 1:
Let Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f(x)={\frac {1}{x^{2}}}} . Since our interval is $[1,5]$ and we are using 4 rectangles, each rectangle has width 1. Since the problem doesn't specify, we can choose either right- or left-endpoints. Choosing left-endpoints, the Riemann sum is
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 1\cdot (f(1)+f(2)+f(3)+f(4))} .

Step 2:
Thus, the left-endpoint Riemann sum is
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 1\cdot (f(1)+f(2)+f(3)+f(4))={\bigg (}1+{\frac {1}{4}}+{\frac {1}{9}}+{1}{16}{\bigg )}={\frac {205}{144}}} .
The left-endpoint Riemann sum overestimates the area of Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle S} .

(b)

Step 1:
Let $n$ be the number of rectangles used in the left-endpoint Riemann sum for Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f(x)={\frac {1}{x^{2}}}} .
The width of each rectangle is Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \Delta x={\frac {5-1}{n}}={\frac {4}{n}}} .

Step 2:
So, the left-endpoint Riemann sum is
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \Delta x{\bigg (}f(1)+f{\bigg (}1+{\frac {4}{n}}{\bigg )}+f{\bigg (}1+2{\frac {4}{n}}{\bigg )}+\ldots +f{\bigg (}1+(n-1){\frac {4}{n}}{\bigg )}{\bigg )}} .
Now, we let $n$ go to infinity to get a limit.
So, the area 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 S} is equal 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 \lim_{n\to\infty} \frac{4}{n}\sum_{i=0}^{n-1}f\bigg(1+i\frac{4}{n}\bigg)} .

Final Answer:
(a) The left-endpoint Riemann sum 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 \frac{205}{144}} , which overestimates the area 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 S} .
(b) Using left-endpoint Riemann sums: 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 \lim_{n\to\infty} \frac{4}{n}\sum_{i=0}^{n-1}f\bigg(1+i\frac{4}{n}\bigg)}

Return to Sample Exam