Cylindrical shells

Question

Use the method of cylindrical shells to find the volume V of the solid obtained by rotating the region bounded by the given curves about the x-axis.
x + y = 7, x = 16 − (y − 3)^2

Answer 1

Hi Pat,

 

In order to find the points of intersection of the line with the parabola, substitute x=7-y for x into the equation of the parabola. You will find that the points of intersection are located at (7,0) and (0,7).

 

Sketch a graph then realize that the height of a typical shell can be represented by 16 - (y-3)² - (7-y) (right x - left x)

 

Take a look here: https://www.desmos.com/calculator/k9u7rg7rty

 

The integral has been set up there, an answer supplied. See if you can complete the work in between.