What this problem is telling us is that every time we make a vertical slice, the slice is the length of the
base of a square. If we want to find the volume of the solid, all we have to do is integrate the area of the
square, from one endpoint of the circle to the other.
The side of the square is the vertical slice whose length is 2y, which we can find by solving the equation
of the circle for y and multiplying by 2. We get y = . Then the length of a side of the square is 2
. Because the area of a square is side^2 , we can find the volume with (16 − 4x^2 ) dx.
Let’s perform the integration, although on some problems you will be permitted to find the answer with a
calculator.