Question

The base of a solid is the circular region bounded by the graph of
x^2 + y^2 =a^2, where a > 0. find the volume of the solid if every cross-section perpendicular to the x-axis is a square?

Answer 1

y = (a^2 - x^2)^.5
1/2 a side of square is y
so area of square at x , A(x) = 4 y^2 = 4 (a^2-x^2)
integrate that dx from x = 0 to x = a and double it for the left half
V = 2 * int(0 to a) (4 a^2 - 4 x^2) dx
= 8 [ a^3 - a^3/3 ] = (16 /3) a^3
check that, I did it fast.