Is this integral set up correctly?

In summary, the task is to find the area within two circles with the equations r=cos(theta) and r=sin(theta) using a double integral. The provided solution involves using Green's Theorem, which can be simplified to either 1/2 \int r^2 d\theta or 2 \int_0^{\frac{1}{2}} \left(\sqrt{(\frac{1}{2})^2-(x-\frac{1}{2})^2}-x \right)dx. Both methods are correct and will yield the same result.
  • #1
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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?
 
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  • #2
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:
  • #3
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.
 

Related to Is this integral set up correctly?

1. How do I know if I set up the integral correctly?

One way to check if you set up the integral correctly is by evaluating it. If the result matches with the given solution or expected value of the integral, then it is set up correctly.

2. What are the common mistakes when setting up an integral?

Common mistakes when setting up an integral include incorrect limits of integration, forgetting to include the differential, and using the wrong formula or method for the integral.

3. Can I use different methods to set up an integral?

Yes, there are different methods for setting up an integral, such as the substitution method, integration by parts, and trigonometric substitution. The method chosen depends on the complexity and type of the integrand.

4. How can I simplify the integral before setting it up?

You can simplify the integral by using algebraic techniques, such as factoring, expanding, or simplifying fractions. This can make the integral easier to set up and evaluate.

5. Is there a specific order in which I should set up an integral?

Yes, the general rule is to integrate the innermost function first and then work your way outwards. This is known as the "reverse order" or "inside-out" approach. However, in some cases, it may be easier to use the "outside-in" approach, depending on the form of the integrand.

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