I had a query about this identity:
$\int{cf(x)dx}$ = c$\int{f(x)dx}$
This might sound like a dumb question but the definite integral is after all an area, so if I multiply it by another constant shouldn't I get a volume, then how is that identity true?
Eg; If I have a square of side length $a$, and a height of $c$, then the area is $a^2$ but if I multiply $c$ to it, it becomes $ca^2$, which is a volume. Therefore, it should work with integrals as well.
Generally speaking, when referring to a constant, it is a number without a unit. In your example of the cube, the height necessarily has a unit, so the equation becomes $(c$ units$)(a$ units$)^2$=$ca^2$ units$^3$. In the example with the integral , $c$ has no units, and so it does not affect the units of the integral. Imagine if I were to tile a floor with 10 squares of 30in$^2$. I would multiply $10*30$in$^2$=$300$in$^2$. Notice how the units did not change because I multiplied by a constant.