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joannaj93
 2 years ago
Using a polar double integral, find the volume in the first octant below the surface z=xy, inside cylinder r=4cos(theta), and between the cylinders x^2 + y^2=4 and x^2 + y^2=8.
joannaj93
 2 years ago
Using a polar double integral, find the volume in the first octant below the surface z=xy, inside cylinder r=4cos(theta), and between the cylinders x^2 + y^2=4 and x^2 + y^2=8.

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TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.2dw:1352751755636:dw

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.2in polar the three graphs are\[r=2\]\[r=2\sqrt2\]\[r=4\cos\theta\]can you find the intersection points in terms of theta?dw:1352752221119:dw

joannaj93
 2 years ago
Best ResponseYou've already chosen the best response.0Make the equations equal to one another...

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.2yes, and what do you get for theta?

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.2well that's one, what angle does that correspond to? and what is the other intersection?

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.2yes, 45 and 60, but you better use radians in calcdw:1352754013710:dw

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.2dw:1352754159926:dw

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.2so which fuunction is the outer radius and which one the inner radius for\[0\le\theta\le\frac\pi4\]?

joannaj93
 2 years ago
Best ResponseYou've already chosen the best response.0r=2 is inner and r=2sqrt2 is outer...

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.2yes, good :) and for the interval\[\frac\pi4\le\theta\le\frac\pi3\]what is it?

joannaj93
 2 years ago
Best ResponseYou've already chosen the best response.0r=4cos (theta) is outer and r=2sqrt2 is inner...

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.2I think you have the inner wrongdw:1352754952806:dw

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.2right and what are your integrands?

joannaj93
 2 years ago
Best ResponseYou've already chosen the best response.0I'm not too sure about that

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.2you are integrating xy sub in x=rcost, y=rsint

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.2above t=theta in case you didn't guess also you need to use dA for polar coordinate

joannaj93
 2 years ago
Best ResponseYou've already chosen the best response.0so it'll be the double integral of r^2 cost sint...

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.2that is the xy part, and what is dA in polar?

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.2yes, forget that extra r and you will be sorry :P so can you write your integrals yet?

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.2do you see that it will be two integrals?

joannaj93
 2 years ago
Best ResponseYou've already chosen the best response.0im getting three rs and thethas

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.2there are two regions to considerdw:1352756391639:dwwhat are the bounds for r and theta on the first integral? what are they on the second? those will be the bounds of your integral.

joannaj93
 2 years ago
Best ResponseYou've already chosen the best response.0r=2 to 2√2 and t= 0 to pi/4 and r=2√2 to 4cosθ and t= 0 to pi/3

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.2on the second region the lower bound on r is not 2√2 as we have already discussed, and the lower bound on theta is not 0, but the upper bound of the previous integral

joannaj93
 2 years ago
Best ResponseYou've already chosen the best response.0Sorry about that. give me a sec

joannaj93
 2 years ago
Best ResponseYou've already chosen the best response.0for the second integral: r=2 to 4cost and t= pi/4 to pi/3

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.2good :) so how 'bout setting up those integrals? (two double intergals)

joannaj93
 2 years ago
Best ResponseYou've already chosen the best response.0\[\int\limits_{0}^{\pi/4}\int\limits\limits_{2}^{2\sqrt{2}}r ^{2}\cos \Theta \sin \Theta r dr d \Theta \]

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.2that's the first one, yes

joannaj93
 2 years ago
Best ResponseYou've already chosen the best response.0\[\int\limits_{\pi/4}^{\pi/3} \int\limits_{2}^{4\cos \Theta} r2\cosΘ\sinΘrdrdΘ\]

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.2yes, good :) then add them together to find the total voume happy integrating

joannaj93
 2 years ago
Best ResponseYou've already chosen the best response.0Oh, God this does not look like fun, but Thanks for the help :)

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.2It's really not that hard if you can just remember to integrate with respect to theta and r separately, but I'm still too lazy to do it. Welcome!
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