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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.
 one year ago
 one year 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.
 one year ago
 one year ago

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TuringTestBest ResponseYou've already chosen the best response.2
dw:1352751755636:dw
 one year ago

TuringTestBest ResponseYou've already chosen the best response.2
in 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
 one year ago

joannaj93Best ResponseYou've already chosen the best response.0
Make the equations equal to one another...
 one year ago

TuringTestBest ResponseYou've already chosen the best response.2
yes, and what do you get for theta?
 one year ago

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

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

TuringTestBest ResponseYou've already chosen the best response.2
dw:1352754159926:dw
 one year ago

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

joannaj93Best ResponseYou've already chosen the best response.0
r=2 is inner and r=2sqrt2 is outer...
 one year ago

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

joannaj93Best ResponseYou've already chosen the best response.0
r=4cos (theta) is outer and r=2sqrt2 is inner...
 one year ago

TuringTestBest ResponseYou've already chosen the best response.2
I think you have the inner wrongdw:1352754952806:dw
 one year ago

TuringTestBest ResponseYou've already chosen the best response.2
right and what are your integrands?
 one year ago

joannaj93Best ResponseYou've already chosen the best response.0
I'm not too sure about that
 one year ago

TuringTestBest ResponseYou've already chosen the best response.2
you are integrating xy sub in x=rcost, y=rsint
 one year ago

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

joannaj93Best ResponseYou've already chosen the best response.0
so it'll be the double integral of r^2 cost sint...
 one year ago

TuringTestBest ResponseYou've already chosen the best response.2
that is the xy part, and what is dA in polar?
 one year ago

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

TuringTestBest ResponseYou've already chosen the best response.2
do you see that it will be two integrals?
 one year ago

joannaj93Best ResponseYou've already chosen the best response.0
im getting three rs and thethas
 one year ago

TuringTestBest ResponseYou've already chosen the best response.2
there 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.
 one year ago

joannaj93Best ResponseYou've already chosen the best response.0
r=2 to 2√2 and t= 0 to pi/4 and r=2√2 to 4cosθ and t= 0 to pi/3
 one year ago

TuringTestBest ResponseYou've already chosen the best response.2
on 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
 one year ago

joannaj93Best ResponseYou've already chosen the best response.0
Sorry about that. give me a sec
 one year ago

joannaj93Best ResponseYou've already chosen the best response.0
for the second integral: r=2 to 4cost and t= pi/4 to pi/3
 one year ago

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

joannaj93Best ResponseYou've already chosen the best response.0
\[\int\limits_{0}^{\pi/4}\int\limits\limits_{2}^{2\sqrt{2}}r ^{2}\cos \Theta \sin \Theta r dr d \Theta \]
 one year ago

TuringTestBest ResponseYou've already chosen the best response.2
that's the first one, yes
 one year ago

joannaj93Best ResponseYou've already chosen the best response.0
\[\int\limits_{\pi/4}^{\pi/3} \int\limits_{2}^{4\cos \Theta} r2\cosΘ\sinΘrdrdΘ\]
 one year ago

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

joannaj93Best ResponseYou've already chosen the best response.0
Oh, God this does not look like fun, but Thanks for the help :)
 one year ago

TuringTestBest ResponseYou've already chosen the best response.2
It'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!
 one year ago
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