# 009B Sample Midterm 2, Problem 1

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Consider the region ${\displaystyle S}$ bounded by ${\displaystyle x=1,x=5,y={\frac {1}{x^{2}}}}$ and the ${\displaystyle x}$-axis.

a) Use four rectangles and a Riemann sum to approximate the area of the region ${\displaystyle S}$. Sketch the region ${\displaystyle S}$ and the rectangles and indicate whether your rectangles overestimate or underestimate the area of ${\displaystyle S}$.
b) Find an expression for the area of the region ${\displaystyle S}$ as a limit. Do not evaluate the limit.

Foundations:

Solution:

(a)

Step 1:
Let ${\displaystyle f(x)={\frac {1}{x^{2}}}}$. Since our interval is ${\displaystyle [1,5]}$ 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
${\displaystyle 1\cdot (f(1)+f(2)+f(3)+f(4))}$.
Step 2:
Thus, the left-endpoint Riemann sum is
${\displaystyle 1\cdot (f(1)+f(2)+f(3)+f(4))={\bigg (}1+{\frac {1}{4}}+{\frac {1}{9}}+{1}{16}{\bigg )}={\frac {205}{144}}}$.
The left-endpoint Riemann sum overestimates the area of ${\displaystyle S}$.

(b)

Step 1:
Let ${\displaystyle n}$ be the number of rectangles used in the left-endpoint Riemann sum for ${\displaystyle f(x)={\frac {1}{x^{2}}}}$.
The width of each rectangle is ${\displaystyle \Delta x={\frac {5-1}{n}}={\frac {4}{n}}}$.
Step 2:
So, the left-endpoint Riemann sum is
${\displaystyle \Delta x{\bigg (}f(1)+f{\bigg (}1+{\frac {4}{n}}{\bigg )}+f{\bigg (}1+2{\frac {4}{n}}{\bigg )}+\ldots +f{\bigg (}1+(n-1){\frac {4}{n}}{\bigg )}{\bigg )}}$.
Now, we let ${\displaystyle n}$ go to infinity to get a limit.
So, the area of ${\displaystyle S}$ is equal to ${\displaystyle \lim _{n\to \infty }{\frac {4}{n}}\sum _{i=0}^{n-1}f{\bigg (}1+i{\frac {4}{n}}{\bigg )}}$.
(a) The left-endpoint Riemann sum is ${\displaystyle {\frac {205}{144}}}$, which overestimates the area of ${\displaystyle S}$.
(b) Using left-endpoint Riemann sums: ${\displaystyle \lim _{n\to \infty }{\frac {4}{n}}\sum _{i=0}^{n-1}f{\bigg (}1+i{\frac {4}{n}}{\bigg )}}$