Volume of revolution and areas

[Liondancer]

I'm having a bit of trouble when it comes to volume of revolutions and areas. I find it quite difficult when it comes to setting up the integral. Could someone explain to me or give me a tutorial on how to set up the equations thanks!

Here are a few examples

The region enclosed by the curves y = x^2 and x = y^2 is rotated about the line y = -2. Find the volume of the resulting solid.

Find the volume of the solid formed by rotating the region enclosed by the curves y=e^(x) + 2, y=0 , x=0, and x=0.1 about the x-axis.

Find the volume of the solid obtained by rotating the region enclosed by the curves y=x^2 and x = y^2 about the line x=-1.

The region enclosed by the curves x = 1 - y^4 and x = 0 is rotated about the line x = 4. Find the volume of the resulting solid.

Thanks for all the help!

 

[Theorem.]

Hi,
Volume and area questions are relatively simple once you come to terms with what's actually going on. I have been working on some tutorials for volume and area problems on my science community website (although it's still very young in development). We plan on developing a very extensive tutorial database but it's going to be a long process.
Click http://www.theoremsociety.com/forums/index.php?showforum=7" [Broken] for the tutorials page.
Also I would check out homework problems that have been asked here. Volume questions seem to reappear quite often on PF.

 

[Leptos]

For the method of rings/disks:
Rotation over vertical line in form of x = f(y)
Rotation over horizontal line in form of y = f(x)
Formula: pi*((outer radius)² - (inner radius)²)

For the method of cylinders/shells:
Rotation over vertical line in form of y = f(x)
Rotation over horizontal line in for of x = f(y)
Formula: pi*radius*height

Remember to draw diagrams. Keep in mind, when you're given a restriction for x = _ that's a vertical line, and y = _ is a horizontal line.