I'm looking for good examples of physical motivation for integrals over scalar field. Here is an example I've found: If you want to know the final temperature of an object that travels through a medium described with a temperature field then you'll need a line integral It appears to me that the final temperature of our object would depend not only on its path (i.e. the image of the curve) but on the speed as well (if it spends a lot of time in an area with low temperature, it won't be reflected in its path but it clearly would be reflected in its final temperature). So it looks like the value of the integral would be parametrization-dependend (but it shouldn't). So I have two questions: Am I right that the temperature example is off? What are some good examples of physical motivation for integrals over scalar field? (If possible, don't assume any knowledge of physics.) Thank you in advance!