How Integrals Capture the Area Under a Curve: From Riemann Sums to Limits

This article will discuss the definition of integrals, the concept of differentiation, and how to use the fundamental theorem of calculus to solve integrals.

Area Under a Curve

Historically, the development of the integral concept was driven by the geometric problem of finding the area of shapes.

For example, consider a function f(x) between a and b; the area under its curve corresponds to the light‑blue shaded region shown below.

We define the area under the function from a to b as the integral.

How can we define the integral concept with a formula?

In elementary geometry we often subdivide a complex shape into many non‑overlapping simple shapes whose areas are easy to compute. By approximating the area under a curve with many rectangles, we can write the area between a and b as the sum of n non‑overlapping rectangles of width Δx and height f(x_i), where x_i is the x‑coordinate of the left‑bottom corner of the i‑th rectangle. These approximations are illustrated below.

Dividing into more rectangles

Rectangles become denser

Definition of Integration

We express this approximation method with the following formula:

∫_a^b f(x)\,dx = \lim_{n\to\infty} \sum_{i=1}^{n} f(x_i)\,\Delta x

where Δx is the width of each small rectangle and x_i is the x‑coordinate of the rectangle's left‑bottom corner.

As shown in the figures, the approximation improves as the number of rectangles increases. When the number of rectangles approaches infinity, we obtain the limiting value of the approximate area.

This limiting value is the integral of the function from a to b.