Is this integral set up correctly? (1 Viewer)

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G01

Homework Helper
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Find the area of the region within both circles;

r=cos(theta) and
r=sin(theta)

using a double integral.

I made the following:

[tex] 2\int^{\pi/2}_{\pi/4} \int^{\cos\theta}_0 r dr d\theta [/tex]

I multiplied by 2 because the area I have is only half of the total area to be found. Is this correct or am i doing something stupid?
 

NateTG

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I'm not sure that it's a particularly good example for using multiple integrals, but you're effectively using Green's Theorem on something that can be readily handled by:
[tex]A = \frac{1}{2} \int r^2 d\theta[/tex]
or, for those who prefer not to deal with polar integrals:
[tex]A=2 \int_0^{\frac{1}{2}} \left(\sqrt{(\frac{1}{2})^2-(x-\frac{1}{2})^2}-x \right)dx[/tex]

I'm fairily sure that your expression will give a numerically correct result.

http://mathworld.wolfram.com/GreensTheorem.html
 
Last edited:

G01

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Thanks Nate, I agree that its probably easier to do this with
[tex]1/2 \int r^2 d\theta [/tex] but our assignment was to do it using a double integral. Thanks for the help. I'll keep working on it and see if i get the right answer.
 

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