Calculating Integral Using Greens Theorem

[boneill3]

Homework Statement

Use greens theorem to calculate.
$\int_{c}(e^{x}+y^{2})dx+(e^{x}+y^{2})dy$

Where c is the region between y=x²y=x

Homework Equations

Greens Theorem

$\int_{c}f(x.y)dx+g(x,y)dy= \int_{R}\int (\frac{\partial g}{\partial x}-\frac{\partial f}{\partial y})dA$

The Attempt at a Solution

$\frac{\partial g}{\partial x}= 2x$
$\frac{\partial g}{\partial x}= 2y$
Calculate the integral

$\int_{0}^{x}\int_{0}^{\sqrt{y}}2x-2y\text{ }dy dx$

$=\frac{x^2}{2}-\frac{4x^{5/2}}{5}$

Does this look right?
regards

 

[xaos]

with f(x,y)=g(x,y)=exp(x)+y*y, dg/dx=exp(x), the second dg/dx is a typo.

if you want the region bounded by y=x^2 and y=x, the inside integral must be from x^2 to x and the outside 0 to 1 with area element dydx, the result needs to be a value rather than a function, just something to get use to with multiple integrals.

 

[boneill3]

Thanks

$\int_{0}^{1}\int_{x}^{x^2}2x-2y\text{ }dy dx$

$=\frac{1}{30}$

With the outside limits of double integrals eg 0 to 1 do they always have to be constants?
regards

 

[HallsofIvy]

If the result is supposed to be a constant, then, yes, the limits of the integral have to be numbers, not variables!

 

[boneill3]

Thanks