# Vector Form Using the Curl (Line Integral ) in Python

There is an example in Vector Calculus from Madsen & Tromba page 435 that states:

Let $$F=(xy^2,y+x)$$.Integrate $$(\nabla \times F)\cdot k$$ over the region in the first quadrant bounded by the curves $$y=x^2$$ and $$y=x.$$

We first compute the Curl :

$$(\nabla \times F) = \left(0,0,\frac{\partial F_{2}}{\partial x} - \frac{\partial F_{1}}{\partial y} \right)=(1-2xy)k$$

Thus, $$(\nabla \times F)\cdot k =(1-2xy)$$

This can be integrated over the given region $$D$$ using an iterated integral as follows:

\begin{align*} \int \int_{D} (\nabla \times F)\cdot k \quad dxdy &= \int_{0}^{1}\int_{x^2}^{x}(1-2xy)\quad dydx\\ &=\int_{0}^{1} [y-xy] |_{x^{2}}^{x} dx \\ &= \int_{0}^{1} [x-x^3-x^2+x^5]\\ &= 1/2-1/4-1/3+1/6=1/12 \end{align*}

Searching in sympy module in order to calculate this example in python I am confused how they implement the line integral.Does someone know how this can be done in Sympy?