  ### Home > PC > Chapter 3 > Lesson 3.2.2 > Problem3-78

3-78.

You want to estimate $\int _ { 2 } ^ { 5 } 3 x ^ { 2 } d x$ (or $A\left(3x^{2}, 2 ≤ x ≤ 5\right)$) by breaking the interval up into six pieces.

1. Write the sigma notation that will find the area using left-endpoint rectangles.

$\text{Width } = \frac{\text{interval}}{\text{rectangles}} = \frac{3}{6} = \frac{1}{2}$

$\frac{1}{2}\displaystyle\sum\limits_{\textit{k}=0}^{5}3(???)^2$

$\frac{1}{2}\displaystyle\sum\limits_{\textit{k}=0}^{5}3(0.5\textit{k} + 2)^2$

2. Modify your sum in part (a) so that it will find the area using right-endpoint rectangles.

What change would you make to the indices ($k=?$ to ?)?

3. Estimate the area using midpoint rectangles.

What change do you make to $\left(0.5k + 2\right)$ so that each rectangle is measured from the middle?

4. How would you use your previous results to estimate the area using trapezoids?

To estimate the area using trapezoids, average the left- and right-endpoint results.

Use the eTool below to visualize this problem.
Click on the link to the right to view the full version of the eTool: PCT 3-78 eTool