What is the Area Between Two Polar Curves?

[Charismaztex]

Homework Statement

Find the area between the two curves:

$$r=2sin(\theta), r=2(1-sin(\theta))$$

Homework Equations

$$A=\frac{1}{2} \int_{\beta}^{\alpha} r^2 d\theta$$

The Attempt at a Solution

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

Thanks in advance
Charismaztex

 

[payumooli]

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

 

[Charismaztex]

Ahhh, got the answer thanks!