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8-140.

Calculate the volume of the solid constructed as follows: The base of the solid is the region is formed by the curve and the line , while the cross-sections perpendicular to the ‑axis are squares. Homework Help ✎

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 -values?

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.