Originally Posted by

**demode** Here's a worked example I don't understand:

Use Green's Theorem to evaluate

$\displaystyle \int_C x^2ydx+xdy$

along the triangular path with verticies at: (0,0), (1,0) and (1,2).

Answer:

$\displaystyle \int_C x^2y dx+xdy = \iint_R [\frac{\partial (x)}{\partial x}-\frac{\partial(x^2y)}{\partial y}]dA= \int^1_0 \int^{2x}_0 (1-x^2)dydx$

$\displaystyle =\int_0^1 (2x-2x^3)dx=[x^2 - \frac{x^4}{2}]^1_0 = \frac{1}{2}

$

Now, I don't understand how they got "$\displaystyle 2x$" as the upper limit for the integral in:

$\displaystyle \int^1_0 \int^{2x}_0 (1-x^2)dydx$

??

I did sketch the region. So, is "2x" the equation of the line from (0,0) to (1,2)? If so, how did they work out this equation for it?