Finding the area between Polar Curves

In summary, the formula for finding the area between two polar curves is ∫(1/2)r²dθ, and the limits of integration can be determined by finding the points of intersection between the curves. This formula can also be used for curves that intersect, but may require splitting the integral into multiple parts. The main difference between finding the area between polar curves and finding the area under a polar curve is that the former involves finding the area between two curves, while the latter involves finding the area enclosed by a single curve and the origin. Graphing calculators can be used to find the area between polar curves, but understanding the concept and formula is important for effective use.
  • #1
jojo13
20
1

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.
 
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  • #2
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
jojo13 said:

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
LCKurtz said:
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
Yes, ##\frac \pi 2## is correct.
 

1. What is the formula for finding the area between two polar curves?

The formula for finding the area between two polar curves is ∫(1/2)r²dθ, where r is the distance from the origin to the curve and θ is the angle of rotation.

2. How do I determine the limits of integration when finding the area between polar curves?

The limits of integration can be determined by finding the points of intersection between the two curves and using those as the lower and upper limits.

3. Can I use the same formula for finding the area between polar curves that intersect?

Yes, the same formula can be used for finding the area between polar curves that intersect. However, you may need to split the integral into multiple parts if there are more than two points of intersection.

4. What is the difference between finding the area between polar curves and finding the area under a polar curve?

Finding the area between polar curves involves finding the area between two curves, while finding the area under a polar curve involves finding the area enclosed by a single curve and the origin.

5. Can I use a graphing calculator to find the area between polar curves?

Yes, many graphing calculators have built-in functions for finding the area between polar curves. However, it is important to understand the concept and formula behind it in order to use the calculator effectively.

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