Question

The region bounded by the given curve is rotated about the specified axis. Find the volume V of the resulting solid by any method.
x2 + (y − 4)2 = 16; about the y-axis

Answer 1

This is the graph

Answer 2

So its a circle with the center at (0,4) and an r = 4

Answer 3

So I went ahead and translated the circle equation in terms of x = plus or minus sqrt(16-(y-4)^2)

Answer 4

But I figured I could integrate from 0 to 4, double the integration in the end, and only use the positive value for it.

Answer 5

So, using shell, I figured I could set it up shell method style, using \[h=\sqrt{16-(y-4)^{2}}\] and \[r=y\] and end up with the integration\[4\pi \int\limits_{0}^{4}y(\sqrt{16-(y-4)^{2}})dy\]is that the right way to set it up?

Answer 6

4/3 pi r^3 it's a sphere

Answer 7

why shells? it rotated around the y axis. how about washers. r = sqrt(16 - (y-4)^2), h = dy.
integrate from y = 0 to 8.

Answer 8

Oh god, I thought I was supposed to rotate it about the x axis, I misread that entire question and now I feel so stupid 0.0

Answer 9

don't

Answer 10

So it's 256pi/3

Answer 11

that's what you should end up with...

if you rotate the entire circle around the y axis, you'd get twice the volume. but that doesn't make sense so really it should only be the volume of the sphere.

Answer 12

Ah, okay, I see. Haha, thanks. XD

Answer 13

you're welcome