Question

Let R be the square region bounded by y = 2, x = 2, and the
x and y-axes. When R is rotated about the x-axis, what is the volume of the
resulting solid? What is the volume of the solid generated by rotating R about
the y-axis?

Answer 1

first, since it's a square, it does not matter which side is used as the rotational axis.

geometrically, we have a cylinder of height=2, radius=2

v = pi * 2^2 * 2 = 8pi

Analytically,

v = Integral(pi y^2 dx)[0,2]
= Integral(4pi dx)[0,2]
= 4pi x[0,2]
= 8pi