Examples of physical motivation for integrals over scalar field?

Question

I'm looking for good examples of physical motivation for integrals over scalar field.

Here is an example I've found (source):

A rescue team follows a path in a danger area where for each position the degree of radiation is defined. Compute the total amount of radiation gathered by the rescue team along the path.

Does this example make sense? To me, it sounds like the total amount of radiation would depend not only on their path (i.e. the image of the curve) but on the speed as well, so it looks like the value of the integral would be parametrization-dependend (but it shouldn't).

So I have two questions:

1. Am I right that the radiation example is off?

2. What are some good examples of physical motivation for integrals over scalar field? (If possible, don't assume any knowledge of physics.)

Answer 1

If I have understood you question, you are asking:

Are there quantities that in physics can be describe by a integral of a scalar field?

Assume you have a point mass moving on a curve $\gamma$ now define on the curve the curvilinear abscissa $s$. You know that is given a velocity field on the curve. How long is the distance $d$ covered by the body in the time interval $[t_1,t_2]$?

$$d=\int_{t_1}^{t_2}v(s)\,\mathrm{d}s$$