Question

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 line y = 5.

Answer 1

draw the region. Using discs (washers), the volume is

v = ∫[0,4] π(R^2-r^2) dx
where R = 5-y and r = 5-3=2
v = ∫[0,4] π((5-(1+√x))^2-(2)^2) dx = 40π/3

Answer 2

using cylinders, we have

v = ∫[1,3] 2πrh dy
where r=5-y and h=x=(y-1)^2
∫[1,3] 2π(5-y)(y-1)^2 dy = 40π/3

Answer 3

Thank you steve.