Question

Find the area of the shaded region:
r=sqrt(cos(2theta))
r=9cos(theta)
These are polar coordinates btw

Answer 1

\[\int _0^{\frac{\pi }{2}}\int _{\sqrt{\cos (2 \theta )}}^{9 \cos (\theta )}rdrd\theta \]

Answer 2

Can you please explain?

Answer 3

we want the radius to go from smaller region to larger region

Answer 4

as for theta, as you can see from picture, we are going form 0 to 90 degree

Answer 5

How could you find that analytically?

Answer 6

I didn't have to use any math

Answer 7

For example, this problem you need to use some math
http://openstudy.com/users/imranmeah91#/users/imranmeah91/updates/4e2ee0c20b8b63299a4252ab

Answer 8

That one I know how to solve, you just set them equal to find the limits of integration and then subtract the areas. How is this one different than that?

Answer 9

I tries to set them equal but it wasn't really possible analytically. You're saying that you don't have to do that?

Answer 10

Here, we don't need any math to solve for limit of integration, it is obvious from picture. If you were integration in rectangular, then there will be some solving required

Answer 11

From the picture,we know where the region begins and ends

Answer 12

I think I get it. How are you going to solve the problem then?

Answer 13

You mean integrate?

Answer 14

yes, what am I going to be integrating?

Answer 15

I don't think I ever learnt about double integrals. What is that?

Answer 16

Also, my book says the answer is 81pi/8 - 1/2, where did the 1/4 come from?

Answer 17

I am not sure where -1/2 come from but I recheck my work and I still get \begin{align} \frac{81\pi}{8}\end{align}

Answer 18

It says the area is:
\[A _{1}-A_{2}=\int\limits_{0}^{\pi/2}(1/2)81\cos^2(\Theta) d \Theta-\int\limits_{\pi/4}^{?}\] (1/2)cos(2theta) d(theta)

That's all one equation

Answer 19

Any thoughts?

Answer 20

What class is it for?

Answer 21

End of Calc II

Answer 22

I am not sure where pi/2 come from?

Answer 23

If we were doing single integration, it would be
\[\int_0^{\frac{\pi }{2}} \left(\frac{81 \cos ^2(\theta )}{2}-\frac{1}{2} \cos (2 \theta )\right) \, d\theta \]