CPM Homework Help : CALC3RD Problem 8-111

Uh-oh! Arlen needs your help. He is trying to write an integral for the volume of square cross-sections perpendicular to the $y$ -axis where the base is the region bounded by the $y$ -axis and $y = x^2$ over the interval $0 ≤ x ≤ 3$ . Help him set up this integral.

Hint:

$\text{volume by cross-sections }= \int_{x=a}^{x=b}(A(x))dx\text{ or }\text{volume by cross-sections }= \int_{y=a}^{y=b}(A(y))dy$

Step 1:

These cross-sections are squares. This area formula is well known.

Step 2:

The function and the bounds must be in terms of $y$ , because the cross-sections are perpendicular to the $y$ -axis.
Rewrite the function in terms of $y$ and determine the $y$ -values that correspond with the given domain.