Question

Calculus 3

What is the area inside the region bounded by the cardioids r=1+sin(theta) and r=1+cos(theta)?

Answer 1

have you determined your thetas of intersection .... intervals of concern

Answer 2

They intersect at pi/4 and the origin.

Answer 3

somehting like this eh

Answer 4

due to symmetry, we only need to determine half of it

Answer 5

Yes, I know what the region looks like, I'm just unsure of how to set up my double integral?

Answer 6

the hard part is the interval\[\int_{\alpha}^{\beta}\int_{0}^{r(t)}drdt\]

Answer 7

Oh, would it be\[2\int\limits_{\pi/4}^{\pi}\int\limits_{0}^{1+\cos \theta}r drd \theta \] ??

Answer 8

im not sure what the theta interval is yet. but yes, that how we would work it

Answer 9

recall if this is a circle
\[\int_{0}^{2\pi}\int_{0}^{R} r drdt\]
\[\int_{0}^{2\pi} \frac12R^2dt\]
\[\int_{0}^{2\pi}\frac12R^2dt\]
\[\frac12R^2~2\pi=\pi R^2\]

Answer 10

so in this case, we could simply reduce it real quickly to
\[\frac22\int_{a}^{b}(1+cos(t))^2~dt\]

Answer 11

Still no clue??

Answer 12

still trying to verify the intersection is all

Answer 13

45 degrees and 225 degrees?

Answer 14

Yes! Ok just figured it out.

\[2\int\limits\limits_{\pi/4}^{5\pi/4}\int\limits\limits_{0}^{1+\cos \theta}r drd \theta\]