Understanding Definite Integrals: From Geometry to the Fundamental Theorem

Area Under a Curve

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

Consider a function f(x) on an interval [a, b]; the area under the curve between a and b is shown as the light‑blue shaded region in the figure below.

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

In elementary geometry, complex shapes are often divided into many non‑overlapping simple shapes whose areas are easy to compute. By approximating the area under a curve with many rectangles, each of width Δx and height f(x_i), we obtain a sum that approaches the true area as the number of rectangles increases.

Using more rectangles improves the approximation, as shown in the subsequent figures.

Definition of Integration

The approximation method can be expressed by the formula:

∫_a^b f(x)dx = lim_{n→∞} Σ_{i=1}^{n} f(x_i)·Δx

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

As the number of rectangles n increases, the sum becomes a better approximation of the area. When n approaches infinity, the sum converges to a limit, which is the definite integral of f(x) from a to b.

This limit value is precisely the integral of the function f(x) over the interval [a, b].