Integrating Polar Curves over Period

[samtouchdown]

Hello. I am having trouble conceptualizing and/or decisively arriving to a conclusion to this question. When finding the area enclosed by a closed polar curve, can't you just integrate over the period over the function, for example: 3 cos (3θ), you would integrate from 0 to 2pi/3? It intuitively seems so but graphically I am integrating where the curve is not there . Thanks in advance for the help.

 

[LCKurtz]

The answer is no. You need to examine the graph. Part of the problem is that r can be negative so the graph isn't where you would expect for a given $\theta$. If you draw the graph of your example, you will find that it is a 3 leaved rose which is completed as $\theta$ goes from $0$ to $\pi$.

 

[Chirag B]

No. It will not always be from 0 to what makes the inside of the trigonometric function $2\pi$. Here, 0 to $\frac{2\pi}{3}$ will only give one loop. Since $cos\left(n\theta\right)$ gives n loops when n is odd, there are three loops, or petals, here. Going from 0 to $2\pi$ will only give you one loop. Thus, you would have to go from 0 to $6\pi$ to get the whole function. But because the inside of the function is $3\theta$, plugging in $2\pi$ will give you the whole thing.