Math Help - How to find the area of a petal of a rose curve odd number of petals?

1. How to find the area of a petal of a rose curve odd number of petals?

I'm trying to find the area of one petal of r=2cos(3θ). I tried to integrate the area of the whole graph and then divide it by 3 but it doesn't seem to work.
When I tried to find points where r=0, I don't know what values of θ to use because I wasn't sure which two angles would give the area of exactly one petal. (I was worried that I might get an area of two petals or two and a half or something like that.)
Can someone help me to approach this problem please?
Thanks!

2. Re: How to find the area of a petal of a rose curve odd number of petals?

Hey LLLLLL.

First trying to find the region for each petal (in terms of theta) and integrate each petal separately and add them up.

You need to do this because if for example you integrate over petals that balance each other out (or something similar), then you won't get the right answer. It's like when you integrate sin(x) over 0 to 2*pi, the pi - 2*pi region cancels the 0 to pi region out.

3. Re: How to find the area of a petal of a rose curve odd number of petals?

You need to follow the graph as theta increases to see what the limits of integration are for "one petal". These curves are tricky because they can be traced out twice as theta goes around the circle once - once with r positive and the other with r negative. But it doesn't always work that way.

Chiro's example of how you can get into trouble is a good one.

- Hollywood

4. Re: How to find the area of a petal of a rose curve odd number of petals?

Originally Posted by chiro
Hey LLLLLL.

First trying to find the region for each petal (in terms of theta) and integrate each petal separately and add them up.

You need to do this because if for example you integrate over petals that balance each other out (or something similar), then you won't get the right answer. It's like when you integrate sin(x) over 0 to 2*pi, the pi - 2*pi region cancels the 0 to pi region out.
Hi Chiro,
Thank you for your reply! I now understand why my method didn't work out!
However I'm still confused on how to integrate each petal separately. Would you please tell me how?
Thank you!

5. Re: How to find the area of a petal of a rose curve odd number of petals?

Originally Posted by hollywood
You need to follow the graph as theta increases to see what the limits of integration are for "one petal". These curves are tricky because they can be traced out twice as theta goes around the circle once - once with r positive and the other with r negative. But it doesn't always work that way.

Chiro's example of how you can get into trouble is a good one.

- Hollywood
Hi Hollywood,
Is tracing the graph on a graphing calculator the only way to find the interval of theta?

6. Re: How to find the area of a petal of a rose curve odd number of petals?

Hint: When is r(theta) positive? When is it negative?

7. Re: How to find the area of a petal of a rose curve odd number of petals?

Hi,
In my opinion, analysis of polar curves is just plain hard. For example, for your 3 petal rose, the area of one leaf is the total area (integral from 0 to pi) divided by 3 by "obvious symmetry". I don't know how to rigorously justify this except to compute the areas and see that it is true.

Some time ago I proved for my self what the period of the polar curve $r=cos(m\theta/n)$ is, where m and n are relatively prime integers. The proof was fairly complicated. I wouldn't belittle a good graphing calculator which can trace such curves. You might try it to determine what the period is.

I've attached a "simple" polar curve. I'd hate to try and find the area of one of the petals without technology.

8. Re: How to find the area of a petal of a rose curve odd number of petals?

The trick is to recognize that theta going from 0 to pi traces out the whole curve. Theta going from pi to 2*pi just traces over the curve again.

As chiro said, it helps to figure out when r is positive and negative.

- Michael