Volume by Revolution and Base (Multiple Part Question)

Question

Find the volume of the solid created by revolving the function y=x²+2 bounded by the lines:

1. X-Axis
2. Y-Axis
3. y = 4

Answer 1

This question does not entirely make sense because the area described is not bound by the x-axis. Also, the axis or rotation was not given. I will assume the axis of rotation is the y-axis. So we have an area bound by the y-axis, the function f(x) = x^2+2 and the line y = 4 being rotated about the y-axis. We must slice this shape horizontally indicating that the differential will be dy. Therefore the equation must be converted in terms of x, or x = sqrt(y-2).

The bounds of integration are that which show the largest and smallest possible y values. Here, the min y = 2, and the max y =4 because of the bound. Start with the basic volume of rotation

pi (Integral sign) R^2 dy

pi (Integral sign 2 to 4) sqrt(y-2)^2 dy

pi (integral sign 2 to 4) (y-2) dy

pi * [y^2/2 - 2y] evaluated from 4 - 2

2pi