# volume by rotation about x axis

• Nov 25th 2008, 08:16 PM
Frostking
volume by rotation about x axis
I need to obtain the volume of the region bound by y = x^2 y = 1 and the y axis by rotating it around the x axis and then about the y axis I could sure use some help if anyone has the time to explain how to attack this. Thank you
• Nov 26th 2008, 05:12 AM
earboth
Quote:

Originally Posted by Frostking
I need to obtain the volume of the region bound by y = x^2 y = 1 and the y axis by

[1] rotating it around the x axis and

[2] then about the y axis.

I could sure use some help if anyone has the time to explain how to attack this. Thank you

to [1]:

The solid is a cylinder with radius r = 1 and height/length = 2 (from x = -1 to x = 1):

$\displaystyle V = \int_{-1}^1\left(\pi\cdot 1^2 - \pi x^2\right)dx$

to [2]:

The solid is a kind of bowl with the parabola as cross-section:

$\displaystyle V=\int_0^1(\pi \cdot (\sqrt{y})^2)dy$

For your confirmation: The volumes are $\displaystyle \dfrac12 \pi$ and $\displaystyle \dfrac43 \pi$ .