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April 9th, 2018, 09:27 AM  #1 
Member Joined: Sep 2011 Posts: 98 Thanks: 1  Change of variables/ Transformations part 2
I am not sure how I should set my u and v expressions into the uv plane for this question. How should I look at the expression to set u and v expressions? 
April 12th, 2018, 01:56 PM  #2  
Math Team Joined: Jan 2015 From: Alabama Posts: 3,261 Thanks: 895  Quote:
To integrate over that disk, take u going from 0 to 2 and v from 0 to $2\pi$. $9x^2+ 4y^2$ becomes $36u^2 \cos^2(v)+ 36u^2 \sin^2(v)= 36u^2$. Finally the Jacobian is $\left\begin{array}{cc} 2\cos(v) & 2u \sin(v) \\ 3\sin(v) & 3u \cos(v)\end{array}\right= 6u \cos^2(v)+ 6u \sin^2(v)= 6u$ so the integral becomes $\int_0^1\int_0^{2\pi} e^{36u^2}6u dudv= 12\pi\int_0^1 ue^{36u^2}du$. To do that last integral let $y= 36u^2$. Last edited by skipjack; April 13th, 2018 at 05:07 AM.  

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