# Math Help - Volume Around y=r With Shells

1. ## Volume Around y=r With Shells

The region R enclosed by the curves $x-4 y=-29$ and $y=\sqrt{x-3 }+8$ is rotated about the line y=7. Use cylindrical shells to find the volume of the resulting solid.

2. Originally Posted by shannon1111
The region R enclosed by the curves $x-4 y=-29$ and $y=\sqrt{x-3 }+8$ is rotated about the line y=7. Use cylindrical shells to find the volume of the resulting solid.
$x = 4y-29$

$x = (y-8)^2 + 3$

$4y-29 = (y-8)^2 + 3$

$0 = y^2 - 20y + 96$

$0 = (y - 8)(y - 12)$

$y = 8$ ... $y = 12$

$V = 2\pi \int_8^{12} (y-7)[(4y-29) - ((y-8)^2 + 3)] \, dy$