How do you use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by #y = 32 - x^2# and #y= x^2# revolved about the x=4?

Question

Emilia Aguilar · Accepted Answer

0

since #y=32-x^2 and y=x^2# meet at # x=4# so there no solid exist when rotated at x=4.

Axel Alvarez · Answer

undefined To use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by $ y = 32 - x^2 $ and $ y = x^2 $ revolved about $ x = 4 $, follow these steps:

1. Determine the limits of integration by finding the points of intersection between the two curves:
   $ 32 - x^2 = x^2 $
   Solve for $ x $ to find the points of intersection.
   
2. Set up the integral for the volume using the formula for cylindrical shells:
   $$ V = \int_{a}^{b} 2\pi x f(x) dx $$
   where $ f(x) $ represents the height of the shell and $ x $ is the distance from the axis of rotation (in this case, $ x = 4 $).

3. Calculate the height of the shell, which is the difference between the upper and lower curves at a given $ x $ value.

4. Integrate the expression from step 2 over the interval determined in step 1 to find the volume of the solid of revolution.