Question

Integral calculus:

Find the area of the region enclosed by the cardiod r=20-20cos(theta)

help appreciated :)

Answer 1

Take the integral with respect to theta from 0 to 2pi?

Answer 2

:) thanks, what about are the limits for r?

Answer 3

That's not it. Isn't the formula for area:
\[\int_{\alpha}^{\beta}{1 \over 2}r^2 \text{d}\theta\]
\[r^2=(20-20\cos\theta)^2=[20(1-\cos\theta)]^2=400(1-\cos\theta)^2\]
\[A=200\int_0^{2\pi}1-2\cos\theta-\cos^2\theta \text{d}\theta\]

Answer 4

i think this question requires \[\int\limits_{\alpha}^{\beta}\int\limits_{r1(\theta)}^{r2(\theta)} f(r,\theta) r drd \theta\] though....

Answer 5

Eek.. My mistake.. Proper formula is actually the A = 1/2 r^2 theta.

Answer 6

yeah seems logical blockcolder....i get very confused with polar form!!...j

Answer 7

Can you find the area now? :D

Answer 8

Something tells me I need to put '1' as the function inside the integral...to find area. Im abit lost!

Answer 9

My lecturers examples of this topic are a little unhelpful :(

Answer 10

Yes, the function in the double integral is indeed 1.

Answer 11

Can I find the area of the half cardiod and use symmetry?

Answer 12

You can. :D

Answer 13

So what would be my limits then (this is where i come unstuck) for theta and r?

Answer 14

For r, the limits are 0 and 20-20cos(theta). For theta, 0 and pi.

Answer 15

okay and then I just muliply by 2..and walah we have area?

Answer 16

Yes. :D

Answer 17

Thank you very much for the help guys, ill give it a go now!

Answer 18

To make up for my bad answer in the first place.. I will give you this example from my textbook. Hope this helps:
https://skitch.com/jrcdude/8w78n/stewart-calculus-early-transcendentals-7th-txtbk.pdf-page-1-032-of-1-356

Answer 19

Haha, thanks Jason- appreciate it!

Answer 20

Took me a little long but I got \[600pi \]

Answer 21

Found a brilliant example for those interested :)

Answer 22

and forgot to paste the link! haha. here it is! http://www.maths.manchester.ac.uk/service/MATH20003/cardioid.pdf