Not sure if it's just me, but I can't see your equations.
Use Green's theorem to compute the area of one petal of the 32-leafed rose defined by .
It may be useful for recall that the area of a region D enclosed by a curve C can be expressed as .
I'm pretty sure that theta is inbetween [0, pi/16] but that's really as far as I have gotten! Extensive help please!!
I'll do it the right way this time:
Use Green's theorem to compute the area of one petal of the 32-leafed rose defined by
It may be useful for recall that the area of a region D enclosed by a curve C can be expressed as A=(1/2)[integral_C xdy - ydx]
I don't know if you understand it now, the word theta is supposed to be the symbol, the second equation is supposed to represent the integral over C of xdy - ydx (a variant of green's theorem).
I guess I'm just confused on how to use GT in this context. Is there a vector field given? If not do we just assume the vector field is just <0,0,0>? If so, Green's Theorem would just simplify to finding the area of a region using a double integral. If this is the case, you should evaluate the double integral in polar coordinates. So, you will have .
This is the only thing I can figure out. I got the r limits from the equation given, and the theta limits by graphing in polar coords on my calculator. If you set your step to , min to 0 and max to then you will see exactly one petal of that function.
This seems to be the only way to do it to me. Like I said, I'm not sure why they ask you to use GT...there might be something I'm missing. Hope this helps
Where C is the contour path, and R is the region enclosed by the contour path.
If I'm not mistaken, using your values for , I end up with the same integral by applying GT to the line integral:
Thanks Chris, that makes sense.
The definition for GT I have just relates the work integral to the double integral over R of curl F dot n d . I guess I just didn't think about finding area over a region as an application of GT...I would have just done in straight away using a double integral in polar coords.
Thanks for expanding my understanding of the application/usefulness of GT!