Evaluate ${\int }_0^5\sqrt{10x-x^2}dx$ using geometric interpretation.

Find the net area and the total area of the region bounded by $y=5\cos x$ and the $x$-axis between $x=0$ and $x=2\pi$.

Find the length of the curve $y={\int }_1^x\sqrt{t^2-1}dt$, where $1\le x\le 3$.

Find the center of mass $(\bar{x},\bar{y})$ of a thin plate occupying the region bounded on the left by $x=y^2$ and on the right by $x=3y$ for $0\le y\le 3$, with density $\delta \left(y\right)=1+y$. Use horizontal strips.

Determine the value of $p$ so that $f\left(x\right)=e^{-x}$ is a probability density function on the interval $[p,p+2]$.