I just began learning integral calculus. While I was learning about line, surface, and volume integrals, I realized that while the line integral and the surface integral were integrating vector functions, the volume integral was integrating scalar functions.
Why does the volume integral integrate scalar functions? Is there an integral that integrates vector functions over a volume?
Line integral: $\int_\vec a^\vec b \vec V\cdot d \vec l $
Surface integral: $\int_{surface} \vec V\cdot d \vec a$
Volume integral: $\int_{volume} T d\tau$
Yes, you can have $\int_{\text{volume}} \vec V \, d\tau.$ For example, the total momentum of a fluid in a region $\Omega$ is given by $\int_\Omega \rho \vec u \, dV,$ where $\rho$ is the density and $\vec u$ is the local velocity of the fluid.