Riemann sum
An approximation of the area under a curve, built from a sum of rectangle areas.
A Riemann sum approximates the area between a curve y=f(x) and the x-axis on [a,b] by slicing the interval into n subintervals of width \Delta x=\frac{b-a}{n}, erecting a rectangle over each one, and summing the rectangles' areas: \sum_{i=1}^{n}f(x_i^*)\,\Delta x. The point x_i^* sampled in each subinterval — its left endpoint, right endpoint, or midpoint — determines whether the sum is called a left, right, or midpoint Riemann sum, and each gives a slightly different approximation.
Examples
A right-endpoint Riemann sum with n=4 rectangles on [0,4] uses x_1^*=1,\,x_2^*=2,\,x_3^*=3,\,x_4^*=4.
Increasing n (thinner rectangles) generally makes a Riemann sum a closer approximation to the true area.