Of vector fields

Here we are trying to compute integrals of the form

$$\int_C\mathbf{F}\cdot\mathrm{d}\mathbf{r} = \int_C \mathbf{... ...eft(\mathbf{r}(t)\right) \cdot \mathbf{r}'(t) \mathrm{d}t,$$

alternately written

$$\int_C P \mathrm{d}x + Q \mathrm{d}y + R \mathrm{d}z,\mbox{ where }\mathbf{F}=P \mathbf{i}+Q \mathbf{j}+R \mathbf{k}.$$

Let's work with the vector field $\mathbf{F}(x,y,z)=\langle -xy^3, xz, yz^2\rangle$. We will integrate along the curve $C$ parameterized by $\langle t^2, t^3, t^4\rangle$ for $0\leq t\leq 1$.