Thursday, 18 July 2019

integration - What's wrong with my double integral for determining the area of an ellipse?



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?



2π04cos2(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.



Ellipse




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=1r=1cos2θ4+sin2θ9.
Ugh! This will give us the integral
122π01cos2θ4+sin2θ9dθ, which can be done, but this is not the right way to do this problem!


No comments:

Post a Comment

real analysis - How to find limhrightarrow0fracsin(ha)h

How to find lim without lhopital rule? I know when I use lhopital I easy get $$ \lim_{h\rightarrow 0}...