How do you use the shell method to set up and evaluate the integral that gives the volume of the solid generated by revolving the plane region #x+y=6# and #x=7-(y-1)^2# rotated about the x-axis?
Since we are revolving the around the x axis using the shell method we will be integrating with respect to y.
To proceed we need to find where the functions intersect
Now set the two functions equal to each other
Our radius will be some value
Our cylinder height is
The integral for the volume is
Integrating we have
Now evaluating we get