### Home > CALC > Chapter 8 > Lesson 8.4.1 > Problem 8-140

Find the volume of the solid constructed as follows: The base of 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. Homework Help ✎

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.

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

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*)

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

The integral:

Put it all together.