What is the Area Between Two Polar Curves?

In summary, the conversation was about finding the area between two curves using polar coordinates. The person had found the points of intersection and calculated the answer to be 8/3-4√3, but their textbook said the answer was 7/3-4√3. They used a formula in Matlab to calculate the area of one lobe and then multiplied it by 2 for both lobes. Finally, they got the correct answer of 7/3-4√3.
  • #1
Charismaztex
45
0

Homework Statement



Find the area between the two curves:

[tex]r=2sin(\theta), r=2(1-sin(\theta))[/tex]

Homework Equations



[tex]
A=\frac{1}{2} \int_{\beta}^{\alpha} r^2 d\theta
[/tex]

The Attempt at a Solution



I've got the points of intersection at [tex](1,\frac{1}{6}\pi) and (1,\frac{5}{6}\pi)[/tex] and worked out the answer to be [tex]\frac{8}{3}-4\sqrt{3}[/tex] using the angles in the above polar co-ordinates as the limits, however my textbook says that the answer is [tex]\frac{7}{3}-4\sqrt{3}[/tex] Is anyone able to confirm which is the correct answer.

Thanks in advance
Charismaztex
 
Physics news on Phys.org
  • #2
your book is right

2sin(t) is a circle
next is a kind of cycloidal
this is the formula i used in matlab
int((2*sin(t))^2,0,pi/6)+int((2-2*sin(t))^2,pi/6,pi/2)
gives area of one lobe
multiply by 2 for both the lobes
 
Last edited:
  • #3
Ahhh, got the answer thanks!
 

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

The formula for finding the area between two polar curves is ∫(r₂² - r₁²) dθ, where r₂ and r₁ are the radii of the two curves at a given angle θ.

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

The limits of integration are determined by finding the points of intersection between the two polar curves. These points will be the starting and ending angles for the integral.

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

No, the area between two polar curves cannot be negative. The integral will always yield a positive value, representing the area enclosed by the two curves.

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

There are some special cases to consider, such as when the two curves intersect at only one point or if one curve is completely contained within the other. In these cases, the area can be found using multiple integrals or by breaking the region into smaller sections.

5. Can the area between two polar curves be found using Cartesian coordinates?

Yes, the area between two polar curves can also be found using Cartesian coordinates by converting the polar equations to rectangular equations and then using the formula for finding the area between two curves in the x-y plane.

Similar threads

  • Calculus and Beyond Homework Help
Replies
2
Views
1K
  • Calculus and Beyond Homework Help
Replies
10
Views
1K
  • Calculus and Beyond Homework Help
Replies
5
Views
1K
  • Calculus and Beyond Homework Help
Replies
1
Views
1K
Replies
7
Views
1K
  • Calculus and Beyond Homework Help
Replies
1
Views
785
  • Calculus and Beyond Homework Help
Replies
1
Views
903
  • Calculus and Beyond Homework Help
Replies
3
Views
742
Replies
5
Views
934
  • Calculus and Beyond Homework Help
Replies
3
Views
488
Back
Top