The Integral: Area Under a Curve
Slice it infinitely thin, then add it all up
Summary
The integral calculates the area under the graph of a function. The intuitive method: split the area into rectangles and add up their areas. The more rectangles you use, the closer you get to the true area; that is the essence of the Riemann sum.
In depth
The integral has two main interpretations: the first is geometric, the area under a curve; the second is about accumulation, the build-up of a quantity (total distance, total work, total charge). In the 17th century Newton and Leibniz showed that the derivative and the integral are linked by the fundamental theorem of calculus: integrating a function and then differentiating it (or the other way round) brings you back to where you started. Riemann (19th century) gave the rigorous definition of the integral: split the interval into n pieces, choose a sample point in each piece, and add up the areas of rectangles whose heights are the function values at those points. As n goes to infinity, this sum converges to the Riemann integral.
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