Question

Define Q as the region bounded by the functions u(y)=y^2/3+1 and v(y)=y between y=2 and y=4. If Q is rotated around the y-axis, what is the volume of the resulting solid?

Answer 1

To find the volume of the solid obtained by rotating the region Q around the y-axis, you can use the disk method.

Step-by-step explanation:

To find the volume of the solid obtained by rotating the region Q around the y-axis, we can use the disk method.

The volume of each disk is given by the formula V = π(R^2 - r^2)h, where R is the outer radius, r is the inner radius, and h is the height of the disk.

In this case, the outer radius is v(y) = y and the inner radius is u(y) = y^(2/3) + 1. The height of each disk is dy.

Therefore, the volume of the resulting solid is the integral of π(v^2 - u^2)dy from y = 2 to y = 4:

V = ∫π( y^2 - (y^(2/3) + 1)^2 )dy, where y ranges from 2 to 4.

Simplifying this integral will give you the volume of the resulting solid.