Question

Consider the solid S described below.
The base of S is a circular disk with radius 2r. Parallel cross-sections perpendicular to the base are squares.
Find the volume V of this solid.

Answer 1

Equation of a circle with radius of 2r
\[x^{2} + y^{2} = 4r^{2}\]
solving for y
\[y = \sqrt{4r^{2} - x^{2}}\]
Each square has side length of 2y
Area of each square is (2y)^2 = 4y^2
Its best to integrate with respect to x, with bounds from -2r to 2r
so change area so its in terms of x
\[4y^{2} = 16r^{2} - 4x^{2}\]
\[V = \int\limits_{-2r}^{2r} 16r^{2} -4x^{2} dx\]