anonymous
  • anonymous
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
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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jamiebookeater
  • jamiebookeater
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anonymous
  • anonymous
This is the graph
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anonymous
  • anonymous
So its a circle with the center at (0,4) and an r = 4
anonymous
  • anonymous
So I went ahead and translated the circle equation in terms of x = plus or minus sqrt(16-(y-4)^2)

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anonymous
  • anonymous
But I figured I could integrate from 0 to 4, double the integration in the end, and only use the positive value for it.
anonymous
  • anonymous
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?
anonymous
  • anonymous
4/3 pi r^3 it's a sphere
anonymous
  • anonymous
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.
anonymous
  • anonymous
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
anonymous
  • anonymous
don't
anonymous
  • anonymous
So it's 256pi/3
anonymous
  • anonymous
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.
anonymous
  • anonymous
Ah, okay, I see. Haha, thanks. XD
anonymous
  • anonymous
you're welcome

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