Find the area of the region bounded by the graph of $f\left(x\right)=x^3+2$ and the $x$-axis on the interval $[0,2]$.

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

Let $f\left(x\right)=x^6-x^2$ on $[-1,1]$. Find (i) the net area and (ii) the area of the region bounded by the graph of $f$ and the $x$-axis on the given interval.

Which of the following expressions calculates the arc length of the curve $y=2e^x$ from $x=0$ to $x=\ln 3$?

Calculate the area of the region bounded by $y=\ln x$, $y=-1$, and $x=0.5$.