anonymous
  • anonymous
Find the volume of the given solid: Bounded by the cylinder y^2+z^2=4 and the planes x = 2y, x=0, z=0 in the first octant. Please explain step by step
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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katieb
  • katieb
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amistre64
  • amistre64
have you drawn a picture of it yet? :)
amistre64
  • amistre64
|dw:1333483988846:dw| if we look down from above onto the xy plane we should see something that looks like this contraption
amistre64
  • amistre64
the xy plane is the z=0 plane as well

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amistre64
  • amistre64
|dw:1333484124012:dw|
amistre64
  • amistre64
this looks to be it to me
amistre64
  • amistre64
|dw:1333484294354:dw|
amistre64
  • amistre64
i see, this contiues to a point where y=x/2 and y=2 meet
amistre64
  • amistre64
i had a brilliantly drawn picture but openstudy cant decide how to actually post when you hit post so it got frozen in cyberspace, doomed to an existence of scattered bits and bytes a spose
amistre64
  • amistre64
|dw:1333484644314:dw|
amistre64
  • amistre64
x moves from 0 to 4 ; since x/2 = 2 when x=4 y moves from x/2 to 2 z moves from 0 to sqrt(y^2-4) \[\Large \int_{x=0}^{x=4}\int_{y=x/2}^{y=2}\int_{z=0}^{z=\sqrt{4-y^2}}\ dz.dy.dx\]
amistre64
  • amistre64
you have to do these by hand?
anonymous
  • anonymous
its a good start. thank you for your help I have to get to my lab class now I'll look at it again when i come back
amistre64
  • amistre64
if got a new idea :) |dw:1333494153737:dw|
amistre64
  • amistre64
if we add up the area of the all the rectangles formed as y moves infinitesimally from 0 to 2; then the height of the rectangle = sqrt(4-y^2) and the length of the rectangle = 2y \[\int_Al*w\ dA = \int_{0}^{2}2y(4-y^2)^{1/2}\ dy\]
amistre64
  • amistre64
all we need to do to integrate that is to borrow a negative to make -2y the derivative of the inside and undo the chain rule. \[\int_{2}^{0}-2y(4-y^2)^{1/2}\ dy\] \[\frac{2}{3}(4-y^2)^{3/2}\] this goes away at 2 so we can just worry about the y=0 part \[\frac{2}{3}4^{3/2}=\frac{16}{3}\]
amistre64
  • amistre64
|dw:1333494688743:dw|
anonymous
  • anonymous
Thank you so much! Very detailed and excellent explanation

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