Conversion Rule: lim(n->infinity) $\frac{1}{n}$ * sum(r=1 to n) f$\frac{r}{n}$ = integral(0 to 1) f(x) dx

Steps:

1. Write the sum as $\frac{1}{n}$ * sum f$\frac{r}{n}$
2. Replace r/n by x and 1/n by dx
3. Limits: when r starts at 1 (or 0), x = 0; when r = n, x = 1
4. Evaluate the definite integral

General form: lim$\frac{1}{n}$ * sum(r=p to q) f$\frac{r}{n}$ = integral(p/n to q/n) ... but with n->inf, lower limit = lim$\frac{p}{n}$, upper = lim$\frac{q}{n}$.

Example: lim [$\frac{1}{n+1}$ + $\frac{1}{n+2}$ + ... + $\frac{1}{3n}$] = lim $\frac{1}{n}$ sum(r=1 to 2n) $\frac{1}{1+r/n}$ = integral(0 to 2) $\frac{dx}{1+x}$ = ln 3.