Single Variable Calculus, Chapter 5, 5.2, Section 5.2, Problem 8

Suppose that $f(x)$ is an increasing function. Estimate $\displaystyle \int^9_3 f(x) dx$ using three equal sub-intervals with (a) right end points, (b) left end points and (c) midpoints. What can you say about your estimates?


$
\begin{array}{|c|c|}
\hline
x & f(x) \\
\hline
\\
3 & -3.4\\
4 & -2.1\\
5 & -0.6\\
6 & 0.3\\
7 & 0.9\\
8 & 1.4\\
9 & 1.8\\
\hline
\end{array}
$


a.) The width of each rectangle is..

$\displaystyle \Delta = \frac{9 - 3}{ 3} = 2$

So, we can evaluate the area at right end points (starting from $x = 5$)

$
\begin{array}{|c|c|}
\hline\\
x & f(x) \\
5 & -0.6 \\
7 & 0.9 \\
9 & 1.8\\
\hline
\end{array}
$

Now, the total area of the rectangle is..

$2 [-0.6 + 0.9 + 1.8] = 4.2$

b.) By evaluating the area at the left end point (starting from $x = 3$)

$
\begin{array}{|c|c|}
\hline\\
x & f(x) \\
3 & -3.4 \\
5 & -0.6 \\
7 & 0.9\\
\hline
\end{array}
$

Now, the total area of the rectangle is..

$2[-3.4 - 0.6 + 0.9] = -6.2$

c.) At midpoint (starting from $x = 4$)

$
\begin{array}{|c|c|}
\hline\\
x & f(x) \\
4 & -2.1 \\
6 & 0.3 \\
8 & 1.4\\
\hline
\end{array}
$

The total area of the rectangle is

$2 [-2.1 + 0.3 + 1.4] = -0.8$

We can say that our estimates is neither over estimates nor under estimate. Although $f(x)$ is increasing, its value started from negative and changes to positive.