Question

Consider the polar curve r = 1− 2 sin. find the area enclosed by the inner loop of the curve.

Answer 1

I know A = integral 1/2 r^2 d theta from a to be, but how do i know what the inner loop area is. This I would just take big loop area - small loop area. I'm using to seeing 2 eq on probs like this.

Answer 2

* r = 1 - 2 sin theta

Answer 3

* I need to know how to figure out whats the equation for the inner loop at least i think.

Answer 4

ok it is the equation of Limacons with inner loop.
in order to find limits for integration solve
0=1-2sin(theta).

Answer 5

wow. thanks for graphing, cool. -1/2 = sin theta. that would be the equation for inner loop?

Answer 6

ok generally there is no equation for inner loop . the general equation is
\[\Large r=a \pm bsin(\theta)\]
if a<b then there is always inner loop
if a>b then there is no inner loop at all. examples are shown below.

Answer 7

okay okay i'm learning. but now if there is no general eq how am i suppose to setup the problem for the big area - small area? or do i just calculate big area / r = 1-2sintheta

Answer 8

thats good question!
you will have to use he same equation r=1-2sin(theta) for area . it will be the limits of theta that will give you the required data.

Answer 9

hmm I kind of see where this is going. Unfortunately I have to step out but I'll definitely come back to this. But you got me going so far.