Question

Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line.
x = 4(square root of (3y)), x = 0, y = 3;
about the y-axis

V=?????

Answer 1

x=4√(3y)
x^2 = 48y

To use discs, note that the radius of each disc is x, and the thickness is dy, so

v = ∫[0,3] πx^2 dy
= 48π∫[0,3] y dy
= 216π

To use shells, each shell has radius x, and height 3-y, so

v = ∫[0,12] 2πx(3-y) dx
= 2π∫[0,12] x(3-x^2/48) dx
= 216π