volume by rotation about x axis

**Frostking** · November 25th 2008, 08:16 PM

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

**earboth** · November 26th 2008, 05:12 AM

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):

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

to [2]:

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

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

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

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