The Shell Method

Question

Use cylindrical shells to find the volume of the solid obtained by rotating the region bounded on the right by the graph of g(y) = 4sqrty and on the left by the y-axis for 0<= y <= 9, about the x-axis. Round your answer to the nearest hundredth position.

Answer 1

The formula for the volume of revolution about the x-axis is:

V = 2π∫_a^byg(y)dy

Given:

g(y) = 4√y

0 ≤ y ≤ 9

So:

V = 2π∫₀⁹y(4√y)dy

   = 8π∫₀⁹y^3/2dy

   = 8π(2/5*y^5/2)₀⁹

   = (16π/5)(9^5/2-0)

   = (16π/5)(243)

   ≈ 2442.9