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lions [1.4K]
3 years ago
15

Evaluate the given integral by making an appropriate change of variables, where R is the region in the first quadrant bounded by

the ellipse 64x2 + 81y2 = 1. $ L=\iint_{R} {\color{red}9} \sin ({\color{red}384} x^{2} + {\color{red}486} y^{2})\,dA $.
Mathematics
1 answer:
Whitepunk [10]3 years ago
5 0

\displaystyle\iint_R\sin(384x^2+486y^2)\,\mathrm dA

Notice that Given that R is an ellipse, consider a conversion to polar coordinates:

\begin{cases}x(r,\theta)=\frac r8\cos\theta\\y(r,\theta)=\frac r9\sin\theta\end{cases}

The Jacobian for this transformation is

J=\begin{bmatrix}\frac18\cos\theta&-\frac r8\sin\theta\\\frac19\sin\theta&\frac r9\cos t\end{bmatrix}

with determinant \det J=\frac r{72}

Then the integral in polar coordinates is

\displaystyle\frac1{72}\int_0^{\pi/2}\int_0^1\sin(6r^2\cos^2t+6r^2\sin^2t)r\,\mathrm dr\,\mathrm d\theta=\int_0^{\pi/2}\int_0^1r\sin(6r^2)\,\mathrm dr\,\mathrm d\theta=\boxed{\frac{\pi\sin^23}{864}}

where you can evaluate the remaining integral by substituting s=6r^2 and \mathrm ds=12r\,\mathrm dr.

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