Question

The expression (1/50)(sqrt(1/50)+sqrt(2/50)+sqrt(3/50)+...+sqrt(50/50) is a Riemann Sum approximation for

Answer 1

$\dfrac1{50}\left(\sqrt{\dfrac1{50}}+\sqrt{\dfrac2{50}}+\cdots+\sqrt{\dfrac{50}{50}}\right)=\displaystyle\sum_{n=1}^{50}\sqrt{\dfrac n{50}}\frac1{50}$

describes a sum of the areas of 50 rectangles, each of width $\dfrac1{50}$, and the $n$th rectangle has height $\sqrt{\dfrac n{50}}$.

$\sqrt{\dfrac n{50}}$ ranges from a small positive number to 1, which means the integration interval must be $[0,1]$. The sum is then the right-endpoint Riemann sum of $\sqrt x$ over the interval with $50$ equally spaced subintervals, so

$\displaystyle\sum_{n=1}^{50}\sqrt{\dfrac n{50}}\frac1{50}\approx\int_0^1\sqrt x\,\mathrm dx$

The sum itself evaluates to roughly 0.676, while the exact value of the integral is $\dfrac23\approx0.667$.