# Volumes of Revolution

• Dec 13th 2010, 02:05 PM
flybynight
Volumes of Revolution
Consider the region bounded by y = x2 + 7, x = 3, the x-axis and the y-axis.

Rotate the region about the y-axis and show how to find the volume of the resulting solid.

Any help? (If anyone could point me to a good tutorial, that would be cool too)

Peter
• Dec 13th 2010, 02:17 PM
Have you drawn the graph? It's pretty straight forward, it's a basic disk method problem. Read this:

Pauls Online Notes : Calculus I - Volumes of Solids of Revolution / Method of Rings

$\pi\int_0^3(x^2+7)^2dx$
• Dec 13th 2010, 03:29 PM
skeeter
Quote:

Originally Posted by flybynight
Consider the region bounded by y = x2 + 7, x = 3, the x-axis and the y-axis.

Rotate the region about the y-axis and show how to find the volume of the resulting solid.

Any help? (If anyone could point me to a good tutorial, that would be cool too)

Peter

method 1 ... cylindrical shells w/r to x

$\displaystyle V = 2\pi \int_0^3 x(x^2+7) \, dx$

method 2 ... lower cylinder (y = 0 to y = 7) + upper washers (y = 7 to y = 16) w/r to y

$\displaystyle V = 63\pi + \pi \int_7^{16} 3^2 - (y-7) \, dy$

method 3 ... large cylinder (from y = 0 to y = 16) - paraboloid from disks (from y = 7 to y = 16)

$\displaystyle 144\pi - \pi \int_7^{16} y - 7 \, dy$
• Dec 13th 2010, 03:31 PM
flybynight
Thanks skeeter. Method 3 is what the prof has.