you have the right idea on all of them however you may want to consider
the shell method on #3
I have a few problems I'm trying to solve and could use some help on
1) y=4-x^2, y=2-x (rotate about the x axis)
So my inner radius is 2-x, my outer radius is 4-x^2
I integrate from x=-1 to x=3 because that's where they intersect.
So I integrate:
pi(4-x^2)^2 - pi(2-x)^2 which comes out to
48pi - pi4^5/5 + 64pi/3
2) y= secx, y=tanx, x=0, x=1
Integrate from 0 to 1, rotate about x axis
Inner radius is tanx, outer is secx.
So I have to integrate
pi(secx)^2 - pi(tanx)^2 = pisec^2x - pitan^2x =
pi integral sec^2x-sec^2x-1 = -pi(x)
3)y=2x, y=0, x = 1 (thus x=y/2)
Rotate about the line x=1
outside radius = 1-y/2
integratee pi(1-y/2)^2 which comes out to 2/3pi
Then I have to do the same shape, but with a different radius (about the line x=2 and limits of integration... so I figure if that one is right then the next one will be too.)