$\int_{c}(xy-y^2)dx-x^3dy$ over the region enclosed by $y=x^2, y=x+2, x=0$

Question

$\int_{c}(xy-y^2)dx-x^3dy$ over the region enclosed by $y=x^2, y=x+2, x=0$

I'd like to use the Green's theorem to solve this, but I am not sure which region I need to take? Am I missing something or is this question not right?

Answer 1

You should ignore the third constraint, $x=0$, since it serves no purpose. The first two constraints already bound a region and the third does nothing to modify that region.

You can parameterize the path in a counter-clockwise direction as follows:

\begin{eqnarray} x(t)&=&\begin{cases} t&\text{ for }-1\le x\le2\\4-t&\text{ for }2<x\le5 \end{cases}\\ y(t)&=&\begin{cases} t^2&\text{ for }-1\le x\le 2\\6-t&\text{ for }2<x\le5 \end{cases} \end{eqnarray} This will result in straightforward but rather messy definite integrals.

Since Green's Theorem applies you can, instead use

$$\int\int_D-3x^2-x+2y\,dxdy$$

but should reverse the order of integration to get

$$ \int_{-1}^2\int_{x^2}^{x+2}-3x^2-x+2y\,dydx $$

But still rather messy.