CPM Homework Help : CALC3RD Problem 8-140

Calculate the volume of the solid constructed as follows: The base of the solid is the region is formed by the curve $y = e^{x^2}$ and the line $y = e$ , while the cross-sections perpendicular to the $y$ -axis are squares.

Graph:

Start by sketching the base of the solid. Notice the $x$ - and $y$ - coordinates where the functions intersect. Also notice the bottom-most point on the solid. Visualize the square cross-sections that will stand up upon this base, making a 3D figure. Each cross-section is perpendicular to the $y$ axis, which means they will be horizontally aligned. At the bottom, near $y = 1$ , the squares will be very small. As you move up, they will gradually get larger. The largest square, at $y = e$ , will have a side length of $2$ .

Step 1:

The bounds: Since the squares are horizontal, the bounds will be $y$ -values. What are the lowest and the highest $y$ -values?

Enclosed shaded region bounded by upward u shaped curve, vertex at the point (0, comma 1), & horizontal line at about 2.7.

Step 2:

The integrand:
Since the squares are horizontal, the integrand must be written in terms of $y$ .
In other words, solve for $x$ .
$y = e^{x^2}$
$\ln(y) = x^2\ln(e)$

$x=\sqrt{\ln(y)}$

Careful! Observe that, because of symmetry across the $y$ -axis, $x$ represents just half of the side-length of each square.

$\text{side }=2\sqrt{\ln(y)}$

Step 3:

The integral:
Put it all together.

$\text{volume by cross-sections }=\int_{y=a}^{y=b}(\text{area})dy$