$U$ is $(-1,1)$ in $\mathbb{R^1}$ and $\phi:U \to \mathbb{R^3}$ is a parametrization of $\phi(U)$. $\phi(x)=(\sin x,\cos x,x)$, $\alpha$ is a differential form with $\displaystyle\alpha=xdy+ydz$.I am going to find $\displaystyle\int_{\phi(U)}\alpha$.I proceeds as follows:
The Jacobian matrix
$$D\phi(x)=\pmatrix{\cos x \\ -\sin x \\ 1}$$
Then
$$\int_{\phi(U)}\alpha=\int_{\phi(U)}-\sin^2(x)+\cos x$$
Then I am confused between two choices of the next step.I do not know whether it should be $\displaystyle\int_{-1}^1-\sin^2(x)+\cos x\sqrt{\det(D^TD)}du$ or $\displaystyle\int_{-1}^{1}-\sin^2(x)+\cos xdu$.
Could someone help?
$$ \int_{\phi(U)} \alpha = \int_U \alpha\circ\phi = \int_{-1}^{1} (x(t) \, y'(t) \, dt + y(t) \, z'(t) \, dt) = \int_{-1}^{1} (-\sin^2(t) + \cos(t)) \, dt $$