Another Polar Coordinates + Integration Question

[Legendre]

I came across this example on the net :

We are integrating over the region that is the area inside of r = 3 + 2 sin θ and outside of r = 2, working in polar coordinates (r,θ).

What is the limits of integration for θ?


# I already know the answer. But I have no idea how to arrive at the solution.

# Do I sketch r = 3 + 2 sin θ and r = 2? r = 2 is just circle centered at origin with radius 2. What about r = 3 + 2 sin θ?

 

[LCKurtz]

Yes, you should plot them. The curves cross where they have equal r values. Set the r's equal and solve for theta. Then, using your picture to see which region is being asked for, set up an integral like this:

$$A = \int_{\theta_0}^{\theta_1} \int_{r_{inner}}^{r_{outer}} r dr d\theta$$

Check your picture and be sure you are integrating in the positive direction for theta.

 

[Legendre]

Yes, you should plot them. The curves cross where they have equal r values. Set the r's equal and solve for theta. Then, using your picture to see which region is being asked for, set up an integral like this:

$$A = \int_{\theta_0}^{\theta_1} \int_{r_{inner}}^{r_{outer}} r dr d\theta$$

Check your picture and be sure you are integrating in the positive direction for theta.

How do I plot r = 3 + 2 sin θ without resorting to a graphing calculator? (and without knowing that equation represents a Limaçon)

 

[remadl700]

Help! I just got to this site because i am looking for help with wind generators but can't figure out how to start a new thead.

can you help or direct me to where I need to go?

 

[LCKurtz]

How do I plot r = 3 + 2 sin θ without resorting to a graphing calculator? (and without knowing that equation represents a Limaçon)

Make a table of values. For every 30 degrees (pi/6) value of theta plot r.

 

[LCKurtz]

Help! I just got to this site because i am looking for help with wind generators but can't figure out how to start a new thead.

can you help or direct me to where I need to go?

At the top of the page click on Physics Forums. Then click on one that looks good for that topic. Then click on the "New Topic" button there at the top.