Volumes of Revolution 1

[danielk]

Hello, please help me answer this question! I know how to do x axis revolutions, but not y. Thank you!

 

[MarkFL]

Hello, and welcome to FMH!

Here's a diagram:



I would use the washer method. The outer radius is:

[MATH]R=\sqrt{x}-(-1)=\sqrt{x}+1[/MATH]
And the inner radius is:

[MATH]r=x^2-(-1)=x^2+1[/MATH]
Now, to determine the limits of integration, we set:

[MATH]\sqrt{x}=x^2[/MATH]
What real solutions to you get?

 

[danielk]

So, a and b are 0 and 1? and V= pi * integral: a=0, b=1 * [sqrt. x =1]^2 - [x^2+1]^2 . * dx

 

[MarkFL]

Yes:

[MATH]V=\pi\int_0^1 \left(\sqrt{x}+1\right)^2-\left(x^2+1\right)^2\,dx[/MATH]
I would expand the two squared binomials in the integrand, combine any like terms, and integrate term by term using the power rule and the FTOC.

 

[danielk]

Great thanks so much sir!

 

[MarkFL]

Carrying out the indicated integration, we then find:

[MATH]V=\frac{29\pi}{30}[/MATH]
If we use the shell method, we find:

[MATH]V=2\pi\int_0^1 (y+1)(\sqrt{y}-y^2)\,dy=\frac{29\pi}{30}\quad\checkmark[/MATH]