Compute $\int_C{x\,\mathrm dz-z\,\mathrm dy+y\,\mathrm dz}$ where $C \subset \mathbb R$ and $\gamma(t)=(\cos t, \sin t, 2t)$ with $t \in [0,2\pi]$

Question

Compute $$\int_{C}{x\,\mathrm dz-z\,\mathrm dy+y\,\mathrm dz}$$ where $C \subset \mathbb R$ and $\gamma(t)=(\cos t, \sin t, 2t)$ with $t \in [0,2\pi]$

The textbook im using does not show much computation examples and i couldn't find any examples online. Not even sure how to start.

Answer 1

Hint:

I suppose that $C$ is a path in $\mathbb{R}^3$ ( not $\mathbb{R}^3$) and $\gamma$ is its parametric equation.

In this case, on the path, we have:

$$ x=\cos t \quad \rightarrow \quad dx=\sin t \;dt $$ $$ y=\sin t \quad \rightarrow \quad dy=-\cos t \;dt $$ $$ z=2t\quad \rightarrow \quad dz=2dt $$

Substituting you find an integral on $t$:

$$ \int_0^{2\pi} (2\cos t +2t\cos t +2 \sin t)dt $$