Find the volume of the solid whose base is a semicircle

Question

Find the volume of the solid whose base is the semicircle y=sqrt(4-x^2) where -2≤x≤2, and the cross sections perpendicular to the x-axis are squares.

Answer 1

There are two ways to do this. One way is to think about the volume of each cross section. Since the volume of a square is just its length squared, and you know the length of one side is y, the volume for each one is just y², or just (4 − x²). So you can set up your integral:

∫ from -2 to 2 of (4 − x²) dx

And then just apply the rules of exponent integration, and you get 4x -x³/3 evaluated from -2 to 2.

Another way you can do it is to use double integrals. (I actually typed this out before I realized this was probably an early Calc 2 problem and went back to write the first part. So if you haven't been introduced to double integrals yet, this will just be a preview of what's coming.)

Because the cross sections are squares, their height will just be the same as their length, so you can describe the surface of this shape as:

z = y = √(4 − x²)

Now you can set it up as the integral

∫∫ z dA = ∫ from -2 to 2 ∫ from 0 to √(4 − x²) of z dy dx

For the inner integral, you are integrating relative to y, so it just becomes y*√(4 − x²) evaluated from 0 to √(4 − x²), which just reduces to (4 − x²). So now you have,

∫ from -2 to 2 of (4 − x²) dx

Just apply the rules of exponent integration, and you get 4x -x³/3 evaluated from -2 to 2.