Green's theorem for finding area.

[mathman44]

Homework Statement

Use GT to find the area of one petal of the 8-leafed rose given by

$$r=17sin(\theta)$$

Recall that the area of a region D enclosed by a curve C can be found by
$$A=1/2\int(xdy - ydx)$$

I calculated it using the parametrization

$$x=rcos(\theta), y=rcos(\theta)$$

And I found a really long integral, evaluated it from 0 to pi/4, and got the correct answer.
Here is my question: apparently, if x is defined as above, and I find

$$dx = -rsin(\theta), dy = rcos(\theta)$$, then the integral

$$A=1/2\int(xdy - ydx)$$ simplifies nicely to $$1/2\int(r^2)d\theta$$. Evaluating this integral again from 0 to pi/4 gives the correct answer.

So... why is it that I can pretend "r" is a constant when I'm evalutating dx and dy, when really, r is dependent on theta just as the x and y parametrizations are?

 

[mathman44]

Anyone, please?

 

[vela]

Why is it that I can pretend "r" is a constant when I'm evalutating dx and dy, when really, r is dependent on theta just as the x and y parametrizations are?

Simple answer: You can't really. If you work it out properly, you'll see the terms proportional to dr cancel out. It's just a coincidence.

 

[mathman44]

Thanks.