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 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.

Foundations:
Link to Riemann sums page

Solution:

(a)

Step 1:
Since our interval is $[1,5]$ and we are using 4 rectangles, each rectangle has width 1. So, the left-endpoint Riemann sum is
$1(f(1)+f(2)+f(3)+f(4))$.

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

(b)

Step 1:

Step 2:

Final Answer:
(a) ${\frac {205}{144}}$
(b)

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