# Find the area of the region.

1. Feb 14, 2007

### rcmango

1. The problem statement, all variables and given/known data

find the area of the region bounded by r = 3 + 2sin(theta)

heres a pic of the intended equations i have so far: http://img263.imageshack.us/img263/8804/untitledpk1.jpg [Broken]

2. Relevant equations

A = 1/2 B integral A r^2 d(theta)

3. The attempt at a solution

2 * 1/2 integral (3 + 2sin(theta))^2 d(theta

this is as far as i have gotten with this, i am not good with integrals, can someone help me with the rest of this please. thanks.

Last edited by a moderator: May 2, 2017
2. Feb 14, 2007

### Gib Z

ok Well basically you need help with this integral:

$$\int^b_a (3 + 2\sin \theta)^2 d\theta$$.

Expand the binomial. (I'm going to ignore the bounds for now, account for them later for me please :P)

$$\int 9 + 12\sin\theta + 4\sin^2 \theta d\theta$$.

Split the integral up. the first 2 parts are very easy, you should get this.

$$9\theta - 12\cos\theta + 4\int \sin^2\theta d\theta$$.

We know that $$\sin^2 \theta = \frac { 1-\cos {2\theta}}{2}$$. Just sub that in, take out the factor of 1/2, split it up again, easy work. GOGOGO!

3. Feb 14, 2007

### HallsofIvy

Staff Emeritus
Becareful when you are doing the part Gib Z did NOT do- determine the limits of integration. What values of $\theta$ will take you exactly once around the boundary?