Calculate $\int_{\Gamma}\omega$ when $\omega = x^2yzdx+xy^2zdy+xyz^2dz$ and $\Gamma$ is the intersection of surfaces $x=1$ and $y^2+z^2=1$

Question

Calculate $\int_{\Gamma}\omega$, when: $\omega = x^2yzdx+xy^2zdy+xyz^2dz$ and $\Gamma$ is the intersection of the surfaces: $x=1$, $y^2+z^2=1$.

Can you help me with that?

At first, I thought of the parametrization $$x=x$$ $$y=r \cos t$$ $$z=r \sin t$$ but I don't think it's a good one, since I don't know how to write the bounds on it. However, I didn't understand very well how it's the procedure when I have the domain the intersection of some things. Can you help me out with this?

Answer 1

You already have it!:

$$\begin{cases}x=1\\{}\\\begin{cases}x=\cos t\\y=\sin t\end{cases}\;,\;\;t\in[0,2\pi]\end{cases}$$