Help With Washer/Shell Method

Question

If possible may I have a visual of the graphs?

1. Use the washer method to set up the integral computing the volume of the solid generated by revolving the region bounded by y = x^2 − 20 and y = −6x + 35 about the line y = −35.

2. Use the shell method to set up the integral computing the volume of the solid generated by revolving the region bounded by x = y^2 − 55 and x = −6y about the line y = 15.

Answer 1

A. Let's find the points of intersection of the given curves

That is solve the equation x² - 20 = - 6x + 35 , which gives x = 5 and x = -11

Outer Radius of Revolution - 6x + 35 - ( -35 ) = - 6x + 70

Inner Radius of Revolution x² - 20 - ( -35 ) = x² +15

Then Volume = π ∫⁵_-11{ [ 6x-70 ]² - [ x² +15 ]² } d x = 85196. 8 π cubic units.

B. Firstly we find the points of intersection of the curves x = -6y and x = y² - 55.

Solving the equation y² - 55 = - 6y for y we get y = -11 and y =5

The radius of revolution is 15 - y

The length of the shell is - 6y -[ y² - 55] = 55 - 6y - y²

Then the volume is V = 2π ∫_-11⁵ ( 15 -y )( 55-6y-y² ) d y = 24576 π cubic units

Answer 2

desmos.com/calculator/ssngfmf019

View this to get some ideas on part 2:

desmos.com/calculator/ttjampnhys