Riemann Sums and the Definite Integral as a Limit

Imagine you want the exact area under the curve f(x) = (1/5)x^2 between x = 2 and x = 6. You can write that area as the definite integral of (1/5)x^2 from 2 to 6. But how is that number actually defined?

Start the way you already know: slice the region into rectangles and add up their areas. A few wide rectangles give a rough guess. More, thinner rectangles hug the curve better, so the guess improves.

Now take the idea to its limit. Let the number of rectangles n grow without bound, so each rectangle's width (4/n) shrinks toward 0. The heights are sampled at points x_i inside each slice, and the total area is the sum of f(x_i) times the width. As n grows, these Riemann sums settle onto one exact value.

That value is the definition of the definite integral: the limit of the Riemann sums as the rectangles become infinitely thin. Every rule you will later learn about integrals is proved by going back to this limit definition.

Suppose you want the area under a curve, but the curve has no simple shape you can measure. Cover the region with rectangles.

To build a left Riemann sum, split the interval into equal pieces. On each piece, stand a rectangle whose height is the function's value at the left edge. Add up all the rectangle areas, and you have an estimate of the area under the curve.

A right Riemann sum works the same way, except each rectangle's height comes from the right edge of its piece. If the function is climbing, left rectangles hide some area while right rectangles stick out above the curve, so one sum lands low and the other lands high.

When you use more rectangles, each one gets skinnier, the gaps and overlaps shrink, and both sums squeeze toward the true area. Push the number of rectangles to infinity and the estimate becomes exact. That is the definite integral, and it is the whole reason left and right sums matter.

The definite integral of a function f from a to b is defined as a limit of Riemann sums. Split [a, b] into n subintervals of equal width Δx = (b - a)/n, pick any sample point x_i in each subinterval, and form the sum of f(x_i) times Δx. The integral is the limit of these sums as n grows without bound.

The function f is called the integrand, the numbers a and b are the limits of integration, and x is the variable of integration. A continuous function is always integrable, meaning this limit exists and does not depend on how the sample points were chosen.

The integral gives the net area between the curve and the x-axis: area above the axis counts positive and area below counts negative. This one definition is the foundation for every later property of integrals, from linearity to the Fundamental Theorem of Calculus.

People have chased the area under curves for over two thousand years. Archimedes estimated areas by filling a shape with smaller and smaller pieces, a strategy now called the method of exhaustion.

You can do the same with rectangles. First, use sigma notation to write long sums compactly: the symbol Σ means add up the terms that follow. Powers of integers come up so often that it helps to know formulas like the sum of the first n integers and the sum of the first n squares.

Then slice the region under a curve into equal-width rectangles, measure each rectangle's area, and add them with a sigma sum. More rectangles means thinner slices and a better estimate. This rectangle-building skill is the warm-up for Riemann sums, where the estimate is pushed to a limit to get the exact area.