Evaluating a Line Integral given r(t)

Question

I've got a line integral to evaluate and I'm a bit lost as to how to work through it.

The integral is given by

$$ \int 3yz \, dx + z^3 \, dy + 3yz^2 \, dz $$ with the curve given by the parametrization $ r(t) = (\sin^2 t - \cos^2 t) \cdot \hat i + \sin t \cdot\hat j + \cos t\cdot \hat k $ for $ 0\leq t \leq 2\pi$, that lies on the surface $x = y^2 - z^2$.

Any hints?

Answer 1

$$dy = \dfrac{dy}{dt} \, dt = \left( \frac d {dt} \sin t \right) \, dt = (\cos t) \, dt$$ and similarly for $dx$ and $dz$. You get $$ \int_0^{2\pi} (\text{some function of $t$}) \, dt. $$

Answer 2

Just do it the straightforward way. Substitute

$$\begin{align} x&=\sin^2 t-\cos^2 t \\ y&=\sin t \\ z&=\cos t \end{align}$$

find what $dx,dy,dz$ are in terms of t and $dt$, and integrate the resulting expression in $t$ and $dt$ with the bounds $0$ and $2\pi$. Ignore the equation of the surface: that is irrelevant here.

If you cannot finish with this hint, let us know just what you were able to do and we can help your further. But you do need to show us your work.