Let *R* be the region bounded by *x*^{2} + *y*^{2} = 1. Write an integral expression for the volume of the solid with base *R*

whose slices perpendicular to the *x*-axis are squares

Answer

The region *R* is the unit circle:

For this solid we slice perpendicular to the *x*-axis and get squares. The slice at position *x* has side-length .

The volume of this slice is

The volume of the entire solid is

We could also find the volume by calculating the volume of half the solid and then multiplying by 2.

Then we get the integral expression

For symmetric objects like the one above, it's easier to find the volume by finding the volume of half the solid and then multiplying by 2. It's pretty likely you'll be asked to finish the problem by evaluating the integral, and it's easier to evaluate an integral if one of the endpoints is 0.