Riemann Sums
To find the area under a curve, the interval $[a, b]$ is cut into $n$ equal pieces and a rectangle (or trapezoid) is placed on each. The total area of the pieces is a Riemann sum: $S_n = \sum_{i=1}^{n} f(x_i^*)\,\Delta x$. The definite integral is the limit of this sum as $n$ goes to infinity.
What to notice
- For an increasing function the left sum stays below the integral and the right sum above it; for a decreasing function ($1/x$) it is the other way round.
- When $n$ doubles, the error of the left and right sums roughly halves, while the error of the midpoint and trapezoid rules drops to about a quarter.
- The midpoint error is about half the trapezoid error, with the opposite sign.