Finding the area between Polar Curves

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Homework Statement



Find the area of the region that consists of all points that lie within the circle r = 1 but outside the polar equation r = cos(2θ)

Homework Equations



A = ∫ 1/2 (r2^2 - r1^2) dθ, where r2 is outer curve and r1 is inner curve.


The Attempt at a Solution



Here is what the graph looks like if you want to see it: http://www.wolframalpha.com/input/?i=r+=+cos(2x)+polar

Ok so first I had to find out the limits for my integral. I set: 1 = cos(2θ) and got that θ = ∏.

Now I made my integral, but instead of going from -∏ to ∏ I went from 0 to ∏ and multiplied it by 2

A = 2 ∫ 1/2 [(1^2) - (cos(2θ)^2)] dθ, from 0 to ∏

A = ∫ [ (1) - (1/2 cos(4θ) +1)] dθ, from 0 to ∏

I solved it and received: ∏ - ∏/2

Which equals ∏/2.

Just want to know if I did it right. The main part I was worried about was the limits for my integral.
 

Answers and Replies

  • #2
epenguin
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I think it's right but IMHO would benefit from being done more simply.

I guess you could take the area of a circle as known.

For the area enclosed by cos(2θ) you could equally well integrate 0 to π/4 and multiply by 8. But simpler still...

In your polar co-ordinate figure it looks quite plausible that the area enclosed by it is half that of the circle, but is not self-evident.

Whereas it is self-evident in. Cartesian co-ordinates. And you could make the argument purely by symmetry without any integrations.
 
Last edited:
  • #3
LCKurtz
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Homework Statement



Find the area of the region that consists of all points that lie within the circle r = 1 but outside the polar equation r = cos(2θ)

Homework Equations



A = ∫ 1/2 (r2^2 - r1^2) dθ, where r2 is outer curve and r1 is inner curve.


The Attempt at a Solution



Here is what the graph looks like if you want to see it: http://www.wolframalpha.com/input/?i=r+=+cos(2x)+polar

Ok so first I had to find out the limits for my integral. I set: 1 = cos(2θ) and got that θ = ∏.

Now I made my integral, but instead of going from -∏ to ∏ I went from 0 to ∏ and multiplied it by 2

A = 2 ∫ 1/2 [(1^2) - (cos(2θ)^2)] dθ, from 0 to ∏

A = ∫ [ (1) - (1/2 cos(4θ) +1)] dθ, from 0 to ∏

I solved it and received: ∏ - ∏/2

Which equals ∏/2.

Just want to know if I did it right. The main part I was worried about was the limits for my integral.

If you look at the graph for ##0<\theta<\pi## of the two figures, it is a bit of a stretch to describe what you have as an outer curve and an inner curve. Of course, squaring makes it all work out OK. I would have just gone from ##0## to ##\frac \pi 4## and multiplied by ##8##.
 
  • #4
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If you look at the graph for ##0<\theta<\pi## of the two figures, it is a bit of a stretch to describe what you have as an outer curve and an inner curve. Of course, squaring makes it all work out OK. I would have just gone from ##0## to ##\frac \pi 4## and multiplied by ##8##.

I see. That's and interesting way of looking at the problem. Thanks for that.

Also, if I'm not mistaken both give the same answer (∏/2) right?
 
  • #5
LCKurtz
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Yes, ##\frac \pi 2## is correct.
 

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