009B Sample Midterm 2, Problem 1

Consider the region $S$ 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 $S$. Sketch the region $S$ and the rectangles and indicate whether your rectangles overestimate or underestimate the area of $S$.

b) Find an expression for the area of the region $S$ as a limit. Do not evaluate the limit.

Approximation of integral with left endpoints is an overestimate.

Foundations:
Recall:
1. The height of each rectangle in the left-hand Riemann sum is given by choosing the left endpoint of the interval.
2. The height of each rectangle in the right-hand Riemann sum is given by choosing the right endpoint of the interval.
3. See the page on Riemann Sums for more information.

Solution:

(a)

Step 1:
Let $f(x)={\frac {1}{x^{2}}}$. 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
$1\cdot (f(1)+f(2)+f(3)+f(4))$.

Step 2:
Thus, the left-endpoint Riemann sum is
$1\cdot (f(1)+f(2)+f(3)+f(4))={\bigg (}1+{\frac {1}{4}}+{\frac {1}{9}}+{1}{16}{\bigg )}={\frac {205}{144}}$.
The left-endpoint Riemann sum overestimates the area of $S$.

(b)

Step 1:
Let $n$ be the number of rectangles used in the left-endpoint Riemann sum for $f(x)={\frac {1}{x^{2}}}$.
The width of each rectangle is $\Delta x={\frac {5-1}{n}}={\frac {4}{n}}$.

Step 2:
So, the left-endpoint Riemann sum is
$\Delta x{\bigg (}f(1)+f{\bigg (}1+{\frac {4}{n}}{\bigg )}+f{\bigg (}1+2{\frac {4}{n}}{\bigg )}+\ldots +f{\bigg (}1+(n-1){\frac {4}{n}}{\bigg )}{\bigg )}$.
Now, we let $n$ go to infinity to get a limit.
So, the area of $S$ is equal to $\lim _{n\to \infty }{\frac {4}{n}}\sum _{i=0}^{n-1}f{\bigg (}1+i{\frac {4}{n}}{\bigg )}\,=\,\lim _{n\to \infty }{\frac {4}{n}}\sum _{i=0}^{n-1}{\bigg (}1+i{\frac {4}{n}}{\bigg )}^{-2}$.

Final Answer:
(a) The left-endpoint Riemann sum is ${\frac {205}{144}}$, which overestimates the area of $S$.
(b) Using left-endpoint Riemann sums:
$\lim _{n\to \infty }{\frac {4}{n}}\sum _{i=0}^{n-1}{\bigg (}1+i{\frac {4}{n}}{\bigg )}^{-2}$

Return to Sample Exam