It is parameterized as follows: x=2cos(θ) and y=3sin(θ).
This is my double integral. It is not evaluating to the right answer. Why?
∫2π0∫√4cos2(θ)+9sin2(θ)0drdθ
I remember from calculus 3 that to get the area for polar coordinates, I just evaluate dr and dθ. I don't see what's wrong with the limits of integration. The radius is from 0 to the formula and the radians are a full revolution.
Answer
In order to include a diagram, I'm turning my comment into an answer. As I said, the θ that appears in this parametrization is NOT the polar coordinates θ. You can see this quite easily if you think about stretching a circle to make the ellipse. I've substituted t in the parametrization and indicated the polar coordinates θ as well.
To do this integral correctly in polar coordinates you must get the polar coordinates equation of the ellipse for starters:
x24+y29=1⟹(rcosθ)24+(rsinθ)29=1⟹r=1√cos2θ4+sin2θ9.
Ugh! This will give us the integral
12∫2π01cos2θ4+sin2θ9dθ, which can be done, but this is not the right way to do this problem!
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