Green's theorem

boneill3

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

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

Where c is the region between y=x²y=x

2. Relevant equations

Greens Theorem

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


3. The attempt at a solution

[latex]\frac{\partial g}{\partial x}= 2x[/latex]
[latex]\frac{\partial g}{\partial x}= 2y[/latex]
Calculate the integral

[latex]\int_{0}^{x}\int_{0}^{\sqrt{y}}2x-2y\text{ }dy dx[/latex]

[latex]=\frac{x^2}{2}-\frac{4x^{5/2}}{5}[/latex]

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

[latex]\int_{0}^{1}\int_{x}^{x^2}2x-2y\text{ }dy dx[/latex]

[latex]=\frac{1}{30}[/latex]

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