With respect to arc length

These integrals are of the form

$$\int_C f(x,y,z) \mathrm{d}s$$

which can be reparameterized as

$$\int_a^b f\left(x(t),y(t),z(t)\right) \sqrt{{\left(\frac{\m... ...{\left(\frac{\mathrm{d}z}{\mathrm{d}t}\right)}^2} \mathrm{d}t$$

or more compactly as

$$\int_a^b f\left(\mathbf{r}(t)\right) \vert\mathbf{r}'(t)\vert \mathrm{d}t.$$

For example, let's start with $f(x,y)=x^2+y^2$. We will integrate along the curve $C$ parameterized by $\langle \cos t, \sin 2t\rangle$ for $0\leq t\leq 1$.