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.