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.

   Step 1(a):

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

   Step 2(a):

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

   Step 3 (a):

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

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

   Hint (b):

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

3. Estimate the area using midpoint rectangles.

   Hint (c):

   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?

   Answer (d):

   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