Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the curves x=y^(1/2), x=0, and y=4 about the x axis.

Answer 1

v = ∫[0,4] 2πrh dy
where r=y and h=2-√y
v = ∫[0,4] 2πy(2-√y) dy = 32π/5

check, using discs

v = ∫[0,2] πr^2 dx
where r=y=x^2
v = ∫[0,2] πx^4 dx = 32π/5