Double integral in polar coordinates: Find area inside r=1 & r=2sin(ϴ)

[norvegas25]

I'm having a trouble on understanding how to find the angles in polar coordinates for things like cardioid, Rose/cleaver, or the limacon.
No need to solve the problem itself, but would be appreciated if you showed med some tips on the angles.



Ex.1:
find the area inside the both circles
r=1 & r=2sin(ϴ)


or


Ex.2:
The area inside the smaller loop of r =1-2sin(ϴ)

 

[Dr.Peterson]

I'm having a trouble on understanding how to find the angles in polar coordinates for things like cardioid, Rose/cleaver, or the limacon.
No need to solve the problem itself, but would be appreciated if you showed me some tips on the angles.

Ex.1:
find the area inside the both circles
r=1 & r=2sin(ϴ)

or

Ex.2:
The area inside the smaller loop of r =1-2sin(ϴ)

Do you mean, how to find the limits of integration? It would be helpful if you showed the parts you can do (and your efforts at everything) so we could have a clearer picture of what is lacking. Each of these problems is different; but ultimately I think you have to sketch a graph first in order to know what is being asked. (What two circles? What smaller loop?)

After graphing, for the first you just have to solve 2sin(ϴ) = 1 to find the points of intersection (where r is the same). For the second, once you see what loop they are talking about, you see that it begins and ends when r=0, so you solve 1-2sin(ϴ) = 0. Sometimes weird things happen, such as a curve intersecting itself, or two curves intersecting, where ϴ is in the opposite direction. It all depends on the problem.