Use the shell method to set up and evaluate the integral that gives the volume of the solid generated by revolving the plane region about the x-axis.

Question

x + y² = 36

Answer 1

Recall that using cylindrical shells to calculate a volume requires that the "strips" we are rotating are parallel to the axis of rotation, and that the expression we will integrate will be the lateral area of a cylinder, which is given by 2πrh.

The graph of the plane region is a horizontally oriented parabola, opening left, with a vertex at (36,0). Because our axis of rotation is horizontal, our strips are too, which means we will be integrating with respect to y. Lastly, because the region is symmetrical about the axis of rotation, we need only revolve the region above the x-axis, so our bounds for y will be 0 and 6.

V = ∫₀⁶2πrhdy = 2π · ∫₀⁶y(36 - y²)dy ...