What does the definite integral of a function represent?

What is the formula for the average value of a function f(x) on the interval [a, b]?

Let ${\int }_0^{3}p\left(x\right)dx=12$, where $p\left(x\right)$ is an even function. Find ${\int }_{-3}^{3}p\left(x\right)dx$.

Consider a right triangle with vertices at $(0,0)$, $(0,c)$, and $(d,0)$, where $c>0$ and $d>0$. What is the average vertical distance from points on the $x$-axis between $x=0$ and $x=d$ to the hypotenuse?

Calculate the average value of the function $f\left(x\right)=e^x$ on the interval $[0,\ln 5]$.