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

An object's velocity in $\text{m/s}$ is described by $v\left(t\right)=4t-t^2$ for $t$ in $[2,7]$. Calculate the average velocity over this interval.

Let an ellipse be defined by $\frac{x^2}{m^2}+\frac{y^2}{n^2}=1$. For a point $(x,y)$ on the ellipse, let $s^2$ be the square of its distance to the center. Find the average value of $s^2$ as $x$ runs from $-m$ to $m$.

Suppose $h\left(x\right)$ is continuous on $[0,5]$. What does the Mean Value Theorem for Integrals guarantee?

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?