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 $y=6x+7$ and $y=x^2$ rotated about the line $y=49$ ?

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Answer 1 · 2015-09-23T00:26:57

Given a choice, I would use washers, but. . .

Here is a (non-interactive) graph using Desmos (desmos.com)
I have included the line $y=49$ in red.

enter image source here

Here is the region using the Socratic grapher. It isn't quite accurate, but you can zoom in and out and drag the graph around

graph{(y-x^2)(y-6x-7) (sqrt(x+1))/(sqrt(x+1))<= 0 [-8.29, 20.21, -6, 8.26]}

$y=6x+7$ if and only if $x=1/6y-7/6$

$y=x^2$ if and only if $x= +-sqrty$

To use shells we take our representative slices horizontally. So the bounds become

For $y=0$ to $y=1$ , $x$ goes from $-sqrty$ to $sqrty$ .

From $y-1$ to $y=49$ , $x$ goes from $1/6y-7/6$ to $sqrty$ .

Throughout the problem, the radius of the cylindrical shell will be $49-y$

So we need to evaluate two integrals:

$2pi int_0^1 (49-y)(sqrty-(-sqrty))dy = 4pi int_0^1 (49y^(1/2)-y^(3/2))dy$

$= 1936/15pi$

And

$2pi int_1^49 (49-y)(sqrty-(1/6y-7/6))dy = 25296/5pi$

The volume is the sum of the two integrals.

Washers

$pi int_-1^7 [(49-x^2)^2-(49-(6x+7))^2]dx = 77824/15pi$

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 y=6x+7y=6x+7 and y=x^2y=x^2 rotated about the line y=49y=49?