What is the area between two polar curves?

[TheRedDevil18]

Homework Statement

Find the area inside one loop of r = 2cos(3 theta) and outside the circle r = 1

Homework Equations

The Attempt at a Solution

I need to clarify something about the limits of integration. I found the intersection of the two curves to be at an angle of pi/9. This is how I setup my integral

A = 2*integral from 0 to pi/9 of 1/2(2cos(3 theta))^2 d theta

Is it correct ?

 

[vela]

Not quite. You haven't used the fact that the area is outside the circle r=1.

 

[TheRedDevil18]

Not quite. You haven't used the fact that the area is outside the circle r=1.

So my expression is for the area inside the circle ?, I'm confused

 

[vela]

No. Why do you think the circle has do to anything with your expression at all?

 

[TheRedDevil18]

Do I have to subtract away the circle ?, I thought by integrating to the point where the two graphs intersect I would get the area outside the circle

 

[vela]

Think about this. Suppose the question asked you to calculate the area inside the circle. How would the integral change? The two curves still intersect at the same points, so using your logic, you'd end up with same integral. Obviously, that can't be right. There's no reason to believe the area inside and outside the circle are the same.

I recommend you rethink the problem starting from the more general formula for the area
$$A = \iint r\,dr\,d\theta,$$ with the appropriate limits, and try to understand where the formula
$$A = \int \frac 12 r^2 \,d\theta$$ comes from. The latter is a special case of the first one, and you need to understand when you can actually use it.

 

[TheRedDevil18]

Ok, so is this expression right ?

A = 2*integral from 0 to pi/9 of (1/2(2cos(3 theta))^2) - 1/2(1)^2 d theta

 

[vela]

How'd you come up with it?

 

[TheRedDevil18]

At the point where they intersect by integrating the flower petal it includes part of the circle, therefore I have to subtract away that part

 

[LCKurtz]

Ok, so is this expression right ?

A = 2*integral from 0 to pi/9 of (1/2(2cos(3 theta))^2) - 1/2(1)^2 d theta

Yes.

 

[TheRedDevil18]

Ok thanks