Solving Change of Variables Homework: Ellipse 9x^2+4y^2=1

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Homework Statement


Make the appropriate change of variables and evaluate \int\int _R\(sin(9x^2+4y^2)}\;dA


Homework Equations



R is bounded by the ellipse 9x^2+4y^2=1

The Attempt at a Solution



I can't figure out what the substitution should be I tried u=9x^2 and v=4y^2 and that didn't work and I don't know where to go from here.
 
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How about u=3x and v=2y? Then switch to polar coordinates in u and v.
 
ohhh... ok thanks. I should have seen that.
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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