## Green's theorem

1. The problem statement, all variables and given/known data

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

2. Relevant equations

Greens Theorem

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

3. 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

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 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.
 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

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