find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line.

x=2=2sqrty, x=0, y=9 about the y axis.

Is this the disk or shells method and how do I set this problem up?

1 answer

first, clear up the meaning of

x=2=2sqrty

If you mean

x = 2+2√y, then since the curve intersects y=9 at (8,9) we have as usual, two ways to solve it. Using shells of thickness dx,

v = ∫[0,8] 2πrh dx
where r = x and h = 9-y = 9-(x/2 - 1)^2
v = 2π∫[0,8] x(9-(x/2-1)^2) dx = 1024/3 π

Of course, you can also use discs of thickness dy, but you have to allow for the hole in the middle for x in [0,2].

v = ∫[0,1] π(R^2-r^2) dy + ∫[1,9] πr^2 dy
where R = 2+2√y and r = 2-2√y
and where r = x = 2+2√y

I'll let you check my math and verify that they are equal.