You don't really need much math to understand this. Imagine that a person is a perfect cube. The amount of rain hitting the top of this cube is constant-- does not depend on the speed of the cube. The amount of rain hitting the front of the cube is proportional to how fast the cube is moving. From this insight, everything else is algebra.
I wonder if the multivariate nature of this problem is throwing you off more than the differential equation aspect of it. If you know multivariable calculus you can think about this problem at a higher level of abstraction (vector fields and flux). The integral you see is the output of the thinking at that higher level of abstraction.
I bet with proper background and practice you could set things like this up rather easily. You just need to learn to think about the problems at the proper abstraction level and then develop some intuition by actually doing lots of problems.
