Question

Find a parametrization of the curve x^(2/3)+y^(2/3)=1 and use Green's theorem to compute the interior. Screenshot of my work attached.

Answer 1

The hint it gives is to modify this parametrization: x(t)=cos(t), y(t)=sin(t).

Answer 2

Ok, so you need to convert it to parametric or rectangular?
Because I cannot understand what you were doing...

Answer 3

I'm trying to find the area of the interior. So I *think*: \[A(S)=\frac{1}{2}\int\limits_{c}F(-y,x)\]

Answer 4

Actually typing that out just gave me something to try out. Just a sec.

Answer 5

Okay you're using Green's thm in reverse

Answer 6

see
http://math.stackexchange.com/questions/91641/find-a-parametrization-of-the-curve-x-frac23-y-frac23-1-and

Answer 7

\[\mathrm{Area}(R) = \iint_R 1\,dA = \int_C x\,dy = \int_0^{2\pi} \cos^3(t) \cdot 3\sin^2(t)\cos(t)\,dt = \frac{3\pi}{8}\]

Answer 8

it uses below parameterization

x = cos^3 t
y = sin^3 t

Answer 9

Looking now...

Answer 10

Brain cramps. :P

Answer 11

I'm a little confused about this line:

"If you can pick out P and Q such that d!/dz-dP/dy =1, the line integral will compute the area of R."

Answer 12

consider the vector filed F = <0, x> and the closed path is same as the given closed curve : x^(2/3)+y^(2/3)=1

Answer 13

Green's thm gives you

\[\iint_R (1-0) dxdy = \int_C 0dx + xdy\]

yes ?

Answer 14

Ok, I'm good to that point.

Answer 15

which is same as



\[\iint_R 1 dxdy = \int_C~x~dy\]

Answer 16

Notice that left hand side is the area interior

Answer 17

Ah! Ok. In other words, we could call dxdy "dA" instead.

Answer 18

ofcourse dA = dxdy

Answer 19

The idea is to evaluate the simple line integral here for calculating the area instead of the complicated double integral on left hand side

Answer 20

Ok, just reading the rest of the post on StackExchange to see if I can tie it all together now.

Answer 21

Okie :) just holler if somehting doesn't make sense

Answer 22

Thank you. You've been a life-saver today. :)

Answer 23

Well, that did it! The only hint we were given on the problem was the image below. Your explanation and links helped a TON. Thanks thanks thanks thanks thanks! :)

Answer 24

np:) yes for that field also( <-y/2, x/2>), the curl evaluates to 1 so can be used for computing area using green's thm