Express a rectangular sum using standard sums and pass to its limit.
Notes and key methods
With n equal subintervals of width h=(b−a)/n, a right-endpoint sum is \(h\sum_{r=1}^n f(a+rh)\). For a continuous function this tends to the definite integral as n increases.
If f is increasing and nonnegative, the left sum is a lower bound and the right sum is an upper bound. Derive the finite expression before taking a limit, and keep the factor h outside the sum.
Check the conditions
The partition and endpoint convention are stated.
Bounds from monotonicity require the appropriate increasing or decreasing behaviour.
Three worked examples
Example 1 · Limits of areas and sums
Consider the area under \(y=x^2+5x\) on \(0\le x\le 3\). For the finite sums use 7 equal subintervals.