Question

Evaluate \[\int\limits_{C}^{} xydy-y ^{2}dx\] where C is the closed curve consisting of the top half of the circle \[x^{2}+y^{2}=4\] and the x-axis segment \[-2 \le x \le 2.\]

Answer 1

Without using Green's theorem, you can parameterize the curve as
\[C=C_i=\begin{cases}
\begin{cases}
x=2\cos t\\
y=2\sin t\\
0\le t\le \pi
\end{cases}&\text{for }i=1
\\\\
\begin{cases}
x=t\\
y=0\\
-2\le t\le2
\end{cases}&\text{for }i=2
\end{cases}\]
Then evaluate the integral as
\[\int_C=\int_{C_1}+\int_{C_2}\]

Answer 2

If you have the luxury of using Green's theorem, you need to find the right partial derivatives.
\[\int_C M(x,y)~dx+N(x,y)~dy=\int\int_D \left(\frac{\partial N}{\partial x}-\frac{\partial M}{\partial y}\right)~dA\]
where \(D\) would be the region bounded by the curve \(C\).

Answer 3

Thanks!