Computing line integral using the line integral of second kind.

Question

I am just beggining with computation of line integrals. I am supposed to compute the integral $$\int\limits_{\Gamma}(2y\;{\rm d}x-x^2\,{\rm d}z)$$, where $\Gamma$ is making part of the cut of surface $z=4-x^2-y^2$ by plane $y=0$ lying above plane $z=0$ orientated in direction of increasing coordinate $x$. How can I compute integrals like this? I imagine I could use the line integral of second kind but I need to parametrize my curve $\Gamma$ first, which is where I am stuck. Or I could maybe use Green theorem in some way. I appreciate any help, thanks a lot!

Answer 1

Okay,

Lets find the curve $\Gamma$.

This curve is found on the plane $y=0$, therefore substituting in the other surface this imposition one finds that the curve is $z=4-x^2$.

The first integrand in the first of teh integral is $y$, but on $\Gamma$ this vanishes becaouse the curve lies on the plane $y=0$. We have solved one half of the problem.

The integrand in the other integral you have is $z$ that on $\Gamma$ takes the value of $z=4-x^2$, hence we only have to compute the following integral

$$\int_{\Gamma}(4-x^2)\,dx$$

What is necessary now is to establish the limits of integration.

Recall that the curve we have is $z=4-x^2$ if the curve must lie over the plane $z=0$ the limits for $x$ are found equation $z=0$, hence $x^2=4$, resulting $x\in(-2,2)$. Therefore the line integral over $\Gamma$ is finally $$\int_{-2}^{2}(4-x^2)\,dx$$