I have the following $1$ form in $\Bbb{R}^3$: $$\eta=(x+z)~dx+(x+y+z)~dy+(x+z)~dz$$ and I want to compute $\int_\alpha \eta$ where $\alpha$ is the line from $(0,0,0)$ to $(1,1,1)$.
My idea was the following. First I found the following parametrization of the line $$\alpha:[0,1]\rightarrow \Bbb{R}^3;~~~t\mapsto (t,t,t)$$ then $$\int_\alpha \eta=\int_0^1 7t ~dt=\frac{7}{2}$$ where I compute $\eta(\alpha)=2t~(t)'+3t~(t)'+2t~(t)'=2t+3t+2t=7t$
but I'm not sure if I did this correctly. Could someone take a look and tell me if this works?
Thanks for your help.