Polar Coordinate Area between two curves

In summary, the problem involves finding the area enclosed by the curves r = √(3)cos(θ) and r = sin(θ) using the formula A = (1/2)∫(R^2 - r^2)dr, where R is the larger function and r is the smaller function over an interval. After visually identifying the area between the curves, the integral is set up as (1/2)∫(0,π/3)sin^2(θ)dθ + (1/2)∫(π/3,π/2)3cos^2(θ)dθ. The limits of integration are determined by the intersections of the two curves and the tangent
  • #1
cryora
51
3

Homework Statement


Find the area enclosed by the curves:
[tex]r=\sqrt(3)cos(\theta)[/tex]
and
[tex]r=sin(\theta)[/tex]

Homework Equations


The area between two polar curves is given by:
[tex]A=(1/2)\int{R^2 - r^2dr}[/tex] where R is the larger function and r is the smaller function over an interval.

The Attempt at a Solution


I set:
[tex]\sqrt{3}cos(\theta)=sin(\theta)[/tex]
[tex]\sqrt(3)=tan(\theta)[/tex]
[tex]\theta=\pi/3,4\pi/3[/tex]

Graphically, I can see that when [tex]0<\theta<\pi/3[/tex] or [tex]4\pi/3<\theta<\pi[/tex] that [tex]sin(\theta)<\sqrt{3}cos(\theta)[/tex]
and when [tex]\pi/2<\theta<4\pi/3[/tex], [tex]\sqrt{3}cos(\theta)<sin(\theta)[/tex]
So it follows that I will have:
[tex]A=(1/2)\int(0,\pi/3){3cos^2(\theta)-sin^(\theta)d\theta+(1/2)\int(\pi/3,4\pi/3){sin^(\theta)-3cos^2(\theta)d\theta+(1/2)\int(4\pi/3,2\pi){3cos^2(\theta)-sin^(\theta)d\theta[/tex]
The numbers separated by commas inside the parenthesis are the limits of integration. Sorry I'm new at this.

I'm just wondering if this is the right way to set it up for a question like this.
 
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  • #2
What you have isn't even close to what you need to do. The first thing you need to do is draw a polar coordinate picture of your two curves and identify visually what area is between the curves. Do you know what the graphs look like?
 
  • #3
Ok, I realize my foolish mistake. So it appears to be two circles, with intersections at (√(3)/2, ∏/3) and the pole. I guess what I do now is integrate once from 0 to ∏/3 using r = sin(θ), and add another integral from ∏/3 to ∏/2 using r = √(3)cos(θ).

So what I'll have is:
[tex]\frac{1}{2}\int_0^\frac{\pi}{3} sin^2(\theta) \,d\theta + \frac{1}{2}\int_\frac{\pi}{3}^\frac{\pi}{2} 3cos^2(\theta) \,d\theta[/tex]

I hope this is correct?
 
Last edited:
  • #4
cryora said:
Ok, I realize my foolish mistake. So it appears to be two circles, with intersections at (√(3)/2, ∏/3) and the pole. I guess what I do now is integrate once from 0 to ∏/3 using r = sin(θ), and add another integral from ∏/3 to ∏/2 using r = √(3)cos(θ).

So what I'll have is:
[tex]\frac{1}{2}\int_0^\frac{\pi}{3} sin^2(\theta) \,d\theta + \frac{1}{2}\int_\frac{\pi}{3}^\frac{\pi}{2} 3cos^2(\theta) \,d\theta[/tex]

I hope this is correct?

Much better. Amazing how a picture helps, eh?
 
  • #5
I am actually confused by this question

[PLAIN]http://img3.imageshack.us/img3/3046/unledze.jpg [Broken]

For his second integral

[tex]\frac{1}{2}\int_\frac{\pi}{3}^\frac{\pi}{2} 3cos^2(\theta) \,d\theta[/tex]

Why is it from pi/3 to pi/2? I don't see it.
 
Last edited by a moderator:
  • #6
flyingpig said:
...
Why is it from pi/3 to pi/2? I don't see it.
Because the larger circle is tangent to the y-axis .
 
  • #7
Oh okay never mind i see it. pi/3 to pi/2 sweeps half of the loop and from 0 to pi/3 sweeps the other half
 

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

The formula for finding the area between two curves in polar coordinates is given by: A = ½ ∫θ1θ2 (r22 - r12) dθ, where θ1 and θ2 are the angles at which the curves intersect and r1 and r2 are the equations of the two curves.

2. How do you determine the limits of integration for finding the area between two curves in polar coordinates?

The limits of integration can be determined by finding the intersection points of the two curves and setting those points as the values for θ1 and θ2 in the formula given in the previous answer.

3. Can the area between two curves in polar coordinates be negative?

No, the area between two curves in polar coordinates is always positive since it is a measurement of the regions bounded by the curves and the origin.

4. Are there any special cases to consider when finding the area between two curves in polar coordinates?

Yes, there are two special cases to consider: when the two curves intersect at the origin and when the curves do not intersect at all. In the first case, the area can be found by using the formula given in the first answer with the limits of integration set to 0. In the second case, the area can be found by finding the areas bounded by each individual curve and subtracting the smaller area from the larger one.

5. How is the area between two curves in polar coordinates related to the Cartesian coordinates?

The area between two curves in polar coordinates can be related to the Cartesian coordinates by converting the polar equations to Cartesian equations and finding the area between the curves using the traditional formula for finding area between two curves in Cartesian coordinates.

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