Question

Evaluate the Integral\[\int\limits_{C}^{}ydx+xdy\] C is the curve\[y=x^3, 0 \le x \le 3\]

Answer 1

Parameterize \(C\) with
\[\begin{cases}
x=t\\
y=t^3\\
0\le t\le3
\end{cases}~~\implies~~\begin{cases}dx=dt\\dy=3t^2~dt\end{cases}\]
Then
\[\int_C y~dx+x~dy=\int_0^3\left(t^3+t(3t^2)\right)~dt\]

Answer 2

I must've done something weird. I thought I did that but I was getting a weird answer. Now I seem to be getting the correct one. Thank You.