Area Between Two Curves

Question

Find the volume of the solid created by revolving the function y = x² + 3 bounded by the line y = 4 revolved about y = 5.

Answer 1

First draw a graph for the enclosed area and note the symmetry about the y-axis. Because it is symetrical, we can evaluate from x = 0 to x = 1 and double the answer. If center of rotation is y = 5, the Rinner = 1 and Router = (5-(x&2+3) = 2-x^2.

2Pi (integral sign from 0 to 1) (Ro^2 - Ri^2) dx

2pi (integral sign from 0 to 1) ((2-x^2)^2 - 1) dx

2pi (integral sign from 0 to 1) (4 - 4x^2 _ 4x^4 - 1) dx

2pi (integral sign from 0 to 1) (2 - 4x^2 _ x^4) dx

apply power rule for integrals yields 2pi*(3x- 4x^3/3 + x^/5) evaluated from 0 to 1

2pi(3 - 4/3 + 1/5)