What is the motivation of weighted integral?

Question

The weighted integral of a function $f:\Omega\rightarrow\mathbb{R}$ is something like $\int_\Omega f(x)w(x)dx$.

What is the motivation of it?

Answer 1

In probability theory, when a random variable $X$ has a pdf $w(x)$, we compute things like $$ \mathbb E[\phi(X)] = \int_{\mathbb R} \phi(x)w(x)\;dx. $$

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If $E$ is a region on $\mathbb R^3$, with mass distributed on it, but not uniformly, we can perhaps specify a density $w(x,y,z)$ for $(x,t,z) \in E$. Then the total mass is $$ \int_{\mathbb R^3} w(x,y,z)\;dx\,dy\,dz $$ and we can compute moments like $$ \int_{\mathbb R^3} x w(x,y,z)\;dx\,dy\,dz\\ \int_{\mathbb R^3} (x^2+z^2) w(x,y,z)\;dx\,dy\,dz $$ We have lots of multi-variable calculus problems like this ... compute pressure exerted by water, where the density varies by depth.