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 $.
1 answer:

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

The Jacobian for this transformation is

with determinant 
Then the integral in polar coordinates is

where you can evaluate the remaining integral by substituting
and
.
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