### 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 ^{ }and the line , while the cross-sections perpendicular to the

*-axis are squares.*

Start by sketching the base of the solid. Notice the - and

*- 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*

*axis, which means they will be horizontally aligned. At the bottom, near*

*, the squares will be very small. As you move up, they will gradually get larger. The largest square, at*

*, will have a side length of*

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

*-value?*

The integrand:

Since the squares are horizontal, the integrand must be written in terms of .

In other words, solve for

*.*

Careful! Observe that, because of symmetry across the

*-axis,*

*represents just half of the side-length of each square.*

The integral:

Put it all together.