# Thread: ASAP help Test in 2days. GReens theorem.

1. ## ASAP help Test in 2days. GReens theorem.

use the Green's Theorem to calculate F dr, where C is the boundary of the region enclosed by y=x^2 and y=cube root of x, and F(x,y)=<x^2-y^3,x^3-y^2>. Please help me.

2. Originally Posted by ottmar0
use the Green's Theorem to calculate F dr, where C is the boundary of the region enclosed by y=x^2 and y=cube root of x, and F(x,y)=<x^2-y^3,x^3-y^2>. Please help me.
Greens theorem tells us that if C is any peicewise simple closed curve then

$\displaystyle \oint _C \vec F \cdot d\vec r=\iint_D \left(\frac{\partial }{\partial x}(x^3-y^2)-\frac{\partial }{\partial x}(x^2-y^3) \right)dA=\iint_D(3x^2+3y^2)dA$

Can you finish from here

3. Originally Posted by ottmar0
use the Green's Theorem to calculate $\displaystyle \int_C F\,dr$, where $\displaystyle C$ is the boundary of the region enclosed by $\displaystyle y=x^2$ and $\displaystyle y=\sqrt[3]{x}$, and $\displaystyle F(x,y)=\langle x^2-y^3,x^3-y^2\rangle$. Please help me.
$\displaystyle F(x,y) = \langle P,Q\rangle = \langle x^2-y^3,x^3-y^2\rangle$

$\displaystyle \frac{\partial Q}{\partial x} = 3x^2$

$\displaystyle \frac{\partial P}{\partial y} = -3y^2$

So, $\displaystyle \int\int_A\left(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right)\,dA = 3\int\int_A(x^2+y^2)\,dA$

Keeping in mind that $\displaystyle \sqrt[3]{x}\geq x^2$ on $\displaystyle [0,1]$, the bounds are:

$\displaystyle x=0...1$
$\displaystyle y=x^2...\sqrt[3]{x}$

So you need to integrate:

$\displaystyle 3\int_0^1\int_{x^2}^{\sqrt[3]{x}}(x^2+y^2)\,dy\,dx$