Why do we subtract the integral of the lower function from the integral of the upper function when finding the area between two curves?

What is the general form of the integral to find the area between two curves f(x) and g(x) over the interval [a, b]?

Compute the area of the region bounded by the graph of $f\left(x\right)=3\sin \left(\frac{x}{6}\right)$ and the $x$-axis on $[0,3\pi ]$.

If the side length of a square cross section is given by f(x) = 2x + 1, what is the area function a(x)?

Why is sketching the solid and its cross sections important when setting up a volume integral for a solid of revolution?

Let $T$ be the region bounded by the graph of $f\left(x\right)=\frac{1}{x+3}$, the $x$-axis, and the vertical lines $x=2$ and $x=6$. Find the volume of the solid formed when $T$ is revolved about the $y$-axis.