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anonymous
 3 years ago
change in variables in double integrals
anonymous
 3 years ago
change in variables in double integrals

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anonymous
 3 years ago
Best ResponseYou've already chosen the best response.0\[\int\limits_{}^{}\int\limits_{}^{}e ^{xy}dA\]

anonymous
 3 years ago
Best ResponseYou've already chosen the best response.0where R is bounded by hyperbolas xy=1 and xy=4 and lines y/x=1 and y/x=3

anonymous
 3 years ago
Best ResponseYou've already chosen the best response.0dw:1353376012156:dw

anonymous
 3 years ago
Best ResponseYou've already chosen the best response.0dw:1353376473982:dw

anonymous
 3 years ago
Best ResponseYou've already chosen the best response.0can this be new region of integration?

zepdrix
 3 years ago
Best ResponseYou've already chosen the best response.1\[\large u=xy \quad v=\frac{y}{x}\] Making this substitution? Hmmm yah I think that will work out. Just need to get your substitutions in terms of x= and y= and then take the jacobian from there.

anonymous
 3 years ago
Best ResponseYou've already chosen the best response.0how did you get that u and v?

anonymous
 3 years ago
Best ResponseYou've already chosen the best response.0you just pick any that you want ?

zepdrix
 3 years ago
Best ResponseYou've already chosen the best response.1I chose u based on the integrand, it simplifies the exponential to something like e^u, which makes it a little easier to deal with. The v was based on the limits of integration, I'm not quite sure if i picked v correctly though :D i still need to check hehe

anonymous
 3 years ago
Best ResponseYou've already chosen the best response.0so use \[\int\limits_{1}^{3}\int\limits_{1}^{4}e ^{u}dudv\]

zepdrix
 3 years ago
Best ResponseYou've already chosen the best response.1Wellllll, if you change variables you have to multiply by the Jacobian as part of your change of differentials. It's a bit messy in this one also :\ hmm

zepdrix
 3 years ago
Best ResponseYou've already chosen the best response.1Do you happen to know what the right answer is suppose to be? XD I want to know if I'm on the right track here lol

zepdrix
 3 years ago
Best ResponseYou've already chosen the best response.1Ok well I'll show you how I did it at least XD No promises that it's correct though lolol

anonymous
 3 years ago
Best ResponseYou've already chosen the best response.0oki thanks for your time

zepdrix
 3 years ago
Best ResponseYou've already chosen the best response.1Hmm so the Jacobian looks like this, it's the determinate of this matrix. When we change from dy dx to the other differentials, we have to multiply by this factor. \[\huge \left\begin{matrix}x_u & x_v \\ y_u & y_v\end{matrix}\right\]

zepdrix
 3 years ago
Best ResponseYou've already chosen the best response.1In order to do so, we need our substitutions written in the form x= and y=

zepdrix
 3 years ago
Best ResponseYou've already chosen the best response.1Starting with this substitution, \[\large u=xy \quad v=\frac{y}{x}\]We'll move some stuff around to get x= and y=.

zepdrix
 3 years ago
Best ResponseYou've already chosen the best response.1\[\large x=\frac{u}{y} \qquad y=vx\]

zepdrix
 3 years ago
Best ResponseYou've already chosen the best response.1\[\large x=\frac{u}{(vx)} \qquad y=v\left(\frac{u}{y}\right)\]

zepdrix
 3 years ago
Best ResponseYou've already chosen the best response.1\[\large x=\sqrt{\frac{u}{v}} \qquad y=\sqrt{uv}\]

zepdrix
 3 years ago
Best ResponseYou've already chosen the best response.1From there, we can apply the Jacobian. Taking the partials and placing them into the matrix.

anonymous
 3 years ago
Best ResponseYou've already chosen the best response.0wow thats gonna be messy

zepdrix
 3 years ago
Best ResponseYou've already chosen the best response.1\[\huge x_u = \frac{1}{2v \sqrt{\frac{u}{v}}}\] Yah they get pretty bad :( They clean up nicely though when you take the determinate :D

anonymous
 3 years ago
Best ResponseYou've already chosen the best response.0but the integrate will be the one that i wrote times the jocubian?

zepdrix
 3 years ago
Best ResponseYou've already chosen the best response.1Yah, it'll be in this form \[\large \int\limits \int\limits e^{xy} dA \rightarrow \qquad \int\limits \int\limits e^u \left\begin{matrix}x_u & x_v \\ y_u & y_v\end{matrix}\right du dv\]

zepdrix
 3 years ago
Best ResponseYou've already chosen the best response.1After calculating the Jacobian, I came up with\[\frac{1+v^2}{4v}\]I may have made mistakes somewhere though :D so make sure you try to do it!! I think it'll take too much time to show all the steps to get it though :3 Hopefully you have somewhat of an understanding of that thingy ma jigger :D

zepdrix
 3 years ago
Best ResponseYou've already chosen the best response.1Keep in mind that you don't have to do the quotient or product rule on any of them, since they're only partials (one of the variables is held constant). Only messy chain rules :)

anonymous
 3 years ago
Best ResponseYou've already chosen the best response.0lol alright thanks a lot for ur help

zepdrix
 3 years ago
Best ResponseYou've already chosen the best response.1\[\huge \int\limits_{u=1}^4 \int\limits_{v=1}^3e^u \frac{1+v^2}{4v}dv \;du\]\[=\huge \int\limits\limits_{u=1}^4 e^u \; du \int\limits\limits_{v=1}^3\frac{1+v^2}{4v} \; dv\]

zepdrix
 3 years ago
Best ResponseYou've already chosen the best response.1So I'm coming up with somethign like that D: pshh i dunno..

zepdrix
 3 years ago
Best ResponseYou've already chosen the best response.1You have evil maf teacher too? :3 assigns bajillion problems? lol

anonymous
 3 years ago
Best ResponseYou've already chosen the best response.0hmmm may be not billion but that kind of problems that take 2hrs to sold one like this one:p
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