Often in my engineering courses, professors just switch out a summation with an integral and change some extensive property to an infinitesimal quantity. This is something I have to accept, but I feel like I've never received rigorous justification for this.
For example, I'm working with products of inertia, defined as: $$I_{xy} = \sum m_{i} (x_{i} y_{i}) ,$$
and I'm presented with the substitution $$\sum m_{i} (x_{i} y_{i}) = \int_{\text{body}}^{} (yx)dm$$
which isn't defined anywhere in my textbook or online. I think I'm ready for an explanation of why this is a mathematically sound process.
EDIT: I did a quick derivation myself, tell me if I did anything hand-wavey given the definition of the integral
! $$\int_{a}^{b} F(x) dx = \lim_{n \to \infty}\sum_{i=1}^{n} f(x_{i}) \Delta x_{i}$$ and $$I_{xy} = \sum m_{i} (x_{i} y_{i}) $$ where yi can be expressed explitily as some $$y_{i} = f(x_{i})$$ under constant linear density define $$\Delta x = \frac{b-a}{n},\,\ and\,\ m_{i} = m/n $$ $$\lambda = \frac{m}{b-a} = \frac{\frac{m}{n}}{\frac{b-a}{n}} = \frac{m_{i}}{\Delta x} $$ $$m_{i} = \frac{m}{b-a} \Delta x $$ we get $$I_{xy} = \sum m_{i} (x_{i} y_{i}) = \frac{m}{b-a} \sum x_{i} f(x_{i}) \Delta x $$ let $$x_i = a + i\cdot \Delta x$$ then we can finally take the limit $$I_{xy} = \lim_{n \to \infty} \sum_{i=1}^{n} (x_{i} y_{i}) = \frac{m}{b-a} \sum x_{i} f(x_{i}) \Delta x \\ I_{xy} = \int_{a}^{b} x\cdot f(x) dx$$
It is not mathematically sound.
What they can claim to be doing is passing to the limit, but that's not just how to do it.