Find the volume of the solid whose base is the circle x^2+y^2=25 and the cross sections perpendicular to the x-axis are triangles whose height and base are equal.

Question

Find the area of the vertical cross section A at the level x=1.

Answer 1

The triangle with base at x has base with length 2y where y is on the circle, so the area A of each triangle is

1/2 (2y)(2y) = 2y^2 = 2(25-x^2)

The volume is just the integral of that from -5 to 5.