Here's your set up.
Just for fun, here's the shells method also:
I understand how to do it and i've gotten every problem right so far but i can't see where my error is in this problem. Can someone point it out? I use a variation of the formula to make it easier on myself. The equations are y=x^2 and y =4x-x^2. They intersect at (0,0) and (2,4). It says to revolve the region to find the volume of the solid about the line y=6. That means that the total volume is pi * integral 6^2 from 0 to 2 with respect to x. Then minus the area of the open area... which would be pi*integral (6-(4x-x^2))^2 from 0 to 2 with respect to x and also subtract pi*integral (6-x^2)^2 from 0 to 2 with respect to x. Where is my error. The correct answer is 64pi/3.