A velocity field is given by $u=x^{2} + y$ and $v= x^{3} + 2y$. Find the circulation around a closed contour which is defined by $x = 1,  y = 0, y = 1$ and $x = 0$. Units of $u$ and $v$ are in $m/s$.

Circulation around a rectangular contour

The closed contour is a square and its corner are at $A(0,0), B(1,0), C(1,1), D(0,1)$.
Circulation is given by

$$\Gamma =\int _{ABCD}udx + vdy$$
$$\Rightarrow\Gamma = \int_{A}^{B}udx + \int_{B}^{C}vdy+\int_{C}^{D}udx+\int_{D}^{A}vdy$$
$$=\int_{0}^{1}\left ( x^{2}+y \right )dx + \int_{0}^{1}\left ( x^{3}+2y \right )dy+\int_{1}^{0}\left ( x^{2}+ y \right )dx + \int_{1}^{0}\left ( x^{3}+2y \right )dy$$ In the first integral, $y=0$, in second integral $x=1$, in the third integral, $y=1$ and in the fourth integral $x=0$.
$$=\int_{0}^{1}\left ( x^{2} \right )dx + \int_{0}^{1}\left ( 1 + 2y \right )dy + \int_{1}^{0}\left ( x^{2} + 1 \right )dx + \int_{1}^{0}\left ( 2y \right ) dy$$
$$=\left | \frac{x^{3}}{3} \right |_{0}^{1}+\left | y+ \frac{2y^{2}}{2} \right |_{0}^{1}+\left | \frac{x^{3}}{3} + x \right |_{1}^{0}+\left | \frac{2y^{2}}{2} \right |_{1}^{0}$$
$$=\left [ \frac{1}{3} \right ]+\left [ 1+1 \right ]+\left [ -\frac{1}{3}-1 \right ]+\left [ -1 \right ]=0$$