Explore how integrals accumulate area beneath a curve. Different Riemann sum methods sample the function at different points within each slice.
Integral calculus studies how quantities accumulate. While differential calculus measures how rapidly something changes at a single instant, integrals measure the total effect of those changes over an interval.
Imagine a curve drawn on a graph representing a function f(x) . A natural question arises: how much area lies between that curve and the horizontal axis between two points a and b ?
Because the boundary is curved, the area cannot be calculated using simple geometric formulas. Instead, mathematicians break the region into many thin rectangles. Each rectangle has a width Δx and a height determined by the function value f(x).
By adding the areas of all rectangles we obtain an approximation of the total region beneath the curve. This sum is called a Riemann sum.
As the rectangles become thinner and more numerous, the approximation improves. In the limit where the width of each rectangle approaches zero, the sum converges to the exact area. This limiting value is defined as the integral.
In the simulator above, the rectangles represent the Riemann sum approximation. Increasing the number of rectangles causes them to align more closely with the curve, gradually revealing the true accumulated area beneath it.
This idea extends far beyond geometry. Integrals allow scientists to compute distance from velocity, total mass from density, electrical charge from charge distribution, and probability from probability density functions.
In essence, an integral answers a simple but powerful question: if a quantity changes continuously across an interval, what is the total amount that results from all those infinitesimal contributions?