91Ó°ÊÓ

Compute the integral of the scalar function or vector field over \(\mathbf{r}(t)=\langle\cos t, \sin t, t\rangle\) for \(0 \leq t \leq \pi\). $$ f(x, y, z)=x y+z $$

Evaluate the line integral. \(\int_{\mathcal{C}}(x-y) d x+(y-z) d y+z d z,\) line segment from (0,0,0) to (1,4,4)

Evaluate the line integral. \(\int_{\mathcal{C}} y^{2} d x+z^{2} d y+\left(1-x^{2}\right) d z,\) quarter of the circle of radius 1 in the \(x z\) -plane with center at the origin in the quadrant \(x \geq 0, z \leq 0,\) oriented counterclockwise when viewed from the positive \(y\) -axis