Area enclosed by cardioid r=1-sin(θ) outside circle r=1

[Lancelot59]

Hi there. The book doesn't have an answer in the back for this problem, so I wanted to know if I was setting everything up correctly.

I need to find the area enclosed by r=1-sin(theta) but outside r=1. This is polar by the way. So a nice simple cardioid and circle. I decided to only do half and then double it, because of the symmetry.

Here is the integral I got.
$$2*\frac{1}{2}[\int_{\frac{-\pi}{2}}^{0}{(1-sin(\theta))^{2} d\theta} - \int_{\frac{-\pi}{2}}^{0}{(1)} d\theta}]$$

My answer was $$\frac{5\pi}{2}+\frac{9}{2}$$. I'm fine with integrating, I'm just wondering if I set it up correctly.

 

[EzimOsai]

I am not that good at calculus,but perhaps if you could send me the complete question i might be able to help. Ezim.

 

[Lancelot59]

That is the complete question. Find the area enclosed by r=1-sin(theta) but outside r=1.

 

[EzimOsai]

you mean there is no diagram or figure?

 

[Lancelot59]

you mean there is no diagram or figure?

Nothing is given. I just drew a quick sketch and checked it on wolfram alpha.

 

[vela]

You set-up looks fine, but your final result isn't correct. Check your work.

 

[Lancelot59]

Oh...5/2 + pi/4 I guess.

 

[vela]

I get 2+pi/4.

 

[Lancelot59]

...This is getting really silly really fast. Well we established that it's just an algebra issue. I'll work it out later. Thanks.

 

[rado5]

yes. I solved it with double integral and I got 2+pi/4.