Find the volume of the solid formed by rotating the region bounded by the graph of y equals 1 plus the square root of x, the y-axis, and the line y = 3 about the x-axis.

Answer 1

using shells of thickness dy,

v = ∫[1,3]2πrh dy
where r = y and h = x = (y-1)^2
v = ∫[1,3] 2πy(y-1)^2 dy = 40π/3

using discs (washers) of thickness dx,

v = ∫[0,4]π(R^2-r^2) dx
where R=3 and r=y=1+√x
v = ∫[0,4]π(9-(1+√x)^2) dx = 40π/3