Derivation of area and volume formulas

An extremely useful application of Riemann sum is the derivation of area and volume formulas for simple shapes in 2-D or 3-D.

Volume of a sphere

Our basic volume problem here uses volumes of revolution for its derivation. We can assert that for a circle of radius $r$, it is given by $y=\sqrt{r^2-x^2}$. We partition $0\le x\le r$ into $n$ intervals of length $\Delta x=\frac{r}{n}$. We use the representative point $x_k^{\ast }=x_k=k\Delta x$, and we computer the volume of a cylindrical disc of radius $y(x_k^{\ast })$ with width $\Delta x$.