Riemann sum

An approximation of the area under a curve by adding up areas of thin rectangles.

A Riemann sum for $f$ on $[a,b]$ partitions the interval into $n$ pieces, picks a sample point in each, and sums the rectangle areas $f(x_i^*)\Delta x$ . Letting $n\to\infty$ (rectangles shrink to zero width) turns the sum into the exact definite integral.

Examples

A left-endpoint Riemann sum with $n=4$ rectangles systematically underestimates the area under an increasing curve.