Find the area between y=x^2 and y=4 in the first quadrant rotated around the y-axis.?

Question

I'm a senior currently taking AP Calculus, and I was wonder if somebody could please help me solve this problem? I know the answer is supposed to be 8π but I just can't get the correct answer and I think all of my work is right. Any help would be great, thanks!

Answer 1

Imagine adding up the volumes of very thin horizontal discs stacked on top of each other. The height of each disc is going to be dy. The radius of each disc is x, so the volume of each disk is pi * x² * dy. Now integrate from y = 0 to y= 4:

Integral (pi * x² * dy) =
Integral (pi * y * dy) =
pi * y²/2 | from 0 to 4 =
pi * 4² /2 = 8 * pi