- #1
LeifEricson
- 11
- 0
Homework Statement
Use an appropriate double integral and the substitution
[tex] y = br\sin \theta \text{\ \ \ } x = ar\cos \theta [/tex]
to calculate the bounded area inside the curve:
[tex] {\left( \frac{x^2}{a^2} + \frac{y^2}{b^2} \right)}^2 = \frac{x^2}{a^2} - \frac{y^2}{b^2} [/tex]
(you can assume that [tex] a,b > 0 [/tex])
Homework Equations
The Attempt at a Solution
I began with the substitution and I got the following representation of the curve:
[tex] r = \sqrt{ \cos^2 \theta - \sin^2 \theta } [/tex].
That means that [tex]0 \leq r \leq \sqrt{ \cos^2 \theta - \sin^2 \theta } [/tex].
Now I have to find from where to where [tex] \theta [/tex] goes. But according to the polar representation I get a very segmented range which is:
[tex] 0 \leq \theta \leq \pi / 4 \text{\ \ \ } 7\pi / 4 \leq \theta \leq 2\pi \text{\ \ \ } 3\pi / 4 \leq \theta \leq 5\pi / 4 [/tex]
And this is a problem because I need [tex] \theta [/tex] to range in a continuous range. So it can be used as a boundaries of an integral.
So what should I do?
Edit:
Fixed substitution. It was x = [tex] br\cos \theta [/tex] instead of what it is now.
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