Question

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.

y = sqrt(x+20)
y = x
y = 0

Answer 1

The shell method requires setting up an integral with cylindrical shells. We take the radius as the distance from the y-axis and the height as the difference between the function values. The integral is then evaluated with the proper limits.

Step-by-step explanation:

The student is asking how to use the shell method to find the volume of a solid of revolution by revolving a region around the x-axis. The equations given are y = sqrt(x+20), y = x, and y = 0. When using the shell method, we consider cylindrical shells with a thickness of dr and integrate to find the volume.

To set up the integral for the shell method, identify the radius of each shell as the distance from the y-axis (since rotating around the x-axis), which will be just x in this scenario. The height of each shell is determined by the difference between the functions y = sqrt(x+20) and y = x. As seen in this scenario, the function y = sqrt(x+20) is always above y = x, so our height will be sqrt(x+20) - x.

Thus, the integral to find the volume V will be:

V = 2π ∫ (radius)(height) dx = 2π ∫ x(sqrt(x+20) - x) dx,

where the limits of the integral are the points of intersection between y = sqrt(x+20) and y = x, which need to be found first.