Many times I have dealt with path and surface integrals of the following form
$$\int_C \mathbf{F}\cdot d\mathbf{r} \,\,\,\,\,\textrm{(path integral)}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\int_S \mathbf{F}\cdot d\mathbf{A} \,\,\,\,\, \textrm{(surface integral)}$$
However, I can't remember ever dealing with path and surface integrals of the following form
$$\int_C f\,\,d\mathbf{r} \,\,\,\,\,\textrm{(path integral)}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\int_S f\,\, d\mathbf{A} \,\,\,\,\, \textrm{(surface integral)}$$
I feel like the dot product present in the first form that I stated is much easier because the integrand is a scalar (this type of integral show up in Stoke's theorem, Divergence theorem, etc.). However, in the second type of integral, we are summing many differential vectors, and I can't remember how to deal with these.
For example, the simple path integral $\int_C d\mathbf{r}$ would look like the following.
With this logic, the integral $\int_C f\,\,d\mathbf{r}$ would be the integral in the picture above, but with each of those little blue arrows scaled by a certain amount.
Sadly, I have no idea how to formally solve an integral of this type, and I can't even come up with a logical simplification (like I did for the path integral) of the surface integral case. I'm guessing for the path integral you could break it up into three different integrals for each vector component, but I'm not too sure.
QUESTION: How does one go about solving these types of path/surface integrals? Could you provide some very simple examples? Are there any 'fundamental theorem of calculus' type simplifications for these types of integrals?
Note that the integrals $\int_C \mathbf{F} \cdot d\mathbf{r}$ and $\int_S \mathbf{F} \cdot d\mathbf{A}$ are real numbers, but the integrals $\int_C f \, d\mathbf{r}$ are vectors. To compute $\int_C f\, d\mathbf{r}$ for instance, parametrize $C$ by $\mathbf{r}(t)$, $a \le t \le b$. Then $\int_C f\, d\mathbf{r} = \int_a^b f(\mathbf{r}(t))\mathbf{\dot{r}}(t)\, dt$. For example, let $C$ be the straight line segment from $(0,0)$ to $(1,1)$. Let $f(x,y) = x + y$. Using the parametrization $\mathbf{r}(t) = \left\langle t,t\right\rangle$, $0 \le t \le 1$. Then $\mathbf{\dot{r}} = \left\langle 1, 1\right\rangle$ and $$\int_C f\, d\mathbf{r} = \int_0^1 2t\left\langle 1,1\right\rangle\, dt = \left(\int_0^1 2t\, dt\right)\left\langle 1, 1\right\rangle = \left\langle 1, 1\right\rangle$$