Build the Area

A Riemann sum approximates the area under a curve by dividing it into thin rectangles. The height of each rectangle is the function value at a sample point, and the width is a fraction of the total interval. As you increase the number of rectangles, the sum converges to the true integral.

Left sums sample the function at the left edge of each rectangle, right sums at the right edge, and midpoint sums at the centre. For most functions, midpoint sums converge faster — you need fewer rectangles to reach the same accuracy.

The accumulation function F(x) = ∫ from a to x of f(t) dt tracks how much area has built up as x moves from a. It is the antiderivative of f(x), connecting integration and differentiation through the Fundamental Theorem of Calculus.

The tool uses the trapezoidal rule with 10,000 sub-intervals, which gives accuracy to about 6 decimal places for smooth functions. This serves as a reference to compare your Riemann sum against.