# Calculate the integral using Green's Theorem

Using Green's theorem I want to calculate $\oint_{\sigma}\left (2xydx+3xy^2dy\right )$, where $\sigma$ is the boundary curve of the quadrangle with vertices $(-2,1)$, $(-2,-3)$, $(1,0)$, $(1,7)$ with positive orientation in relation to the quadrangle.



I have done the following:

We consider the space $D=\{(x,y)\mid -2\leq x\leq 1, x-1\leq y\leq x+6\}$.

From Green's Theorem we get the following: \begin{align*}\oint_{\sigma}\left (2xydx+3xy^2dy\right )&=\iint_D\left (\frac{\partial{(3xy^2)}}{\partial{x}}-\frac{\partial{(2xy)}}{\partial{y}}\right )dxdy \\ & = \iint_D\left (3y^2-2x\right )dxdy \\ & = \int_{-2}^1 \int_{x-1}^{x+6}\left (3y^2-2x\right )dydx \\ & = \int_{-2}^1\left (21x^2+91x+217\right )dx \\ & = \frac{1155}{2}=577,5\end{align*}

At the given solution the result should be $332,25$. What have I done wrong?

• The line connecting $\;(-2,1)\,,\,(1,7)\;$ is $\;y=2x+5\;$ , not $\;y=x+6\;$ ... – DonAntonio Nov 28 '17 at 10:00

## 1 Answer

The integral must be

$$\int_{-2}^1\int_{x-1}^{2x+5}(3y^2-2x)\,dy\,dx=\int_{-2}^1\left(y^3-2xy\right)_{x-1}^{2x+5}dx=$$

$$=\int_{-2}^1\left[(2x+5)^3-(x-1)^3-x(2x+5)^2+x(x-1)^2\right]dx=$$

$$=\int_{-2}^1\left[(x+5)(2x+5)^2+(x-1)^2\right]dx=\ldots$$

• Ah yes! Thank you so much! :-) – Mary Star Nov 28 '17 at 11:29