Why the Definite Integral Represents the Area Under the Curve

The definite integral is a powerful concept in calculus that is often interpreted as the area under the curve of a function. This interpretation is rooted in fundamental concepts of calculus and geometry. In this article, we will explore the reasons behind this interpretation.

Concept of Accumulation

The definite integral calculates the accumulation of quantities, which can be visualized as the area under a curve. For a function f(x) defined on the interval [a, b], the integral sums up the infinitesimally small areas of rectangles that lie under the curve from a to b. This concept is essential to understanding the geometric and physical significance of the integral.

Riemann Sums

To define the definite integral, we often use Riemann sums, a method to approximate the area under a curve. This involves the following steps:

Dividing the interval [a, b] into n subintervals of equal width Delta;x. Selecting a sample point x_i* in each subinterval. Calculating the sum of the areas of rectangles formed by these sample points:

Area ≈ ∑_{i1}^{n} f(x_i*) Δx

As n approaches infinity and Delta;x approaches zero, this sum converges to the definite integral:

∫_a^b f(x) dx

Geometric Interpretation

When visualizing the graph of f(x):

The area under the curve from a to b is the total area between the curve and the x-axis. If f(x) is positive, the area is above the x-axis. If f(x) is negative, the area is below the x-axis, and the integral will yield a negative value for that region.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus connects differentiation and integration, stating that if F(x) is an antiderivative of f(x), then:

∫_a^b f(x) dx  F(b) - F(a)

This theorem reinforces the idea that integration accumulates values of f(x) over the interval, yielding a net area that reflects the total accumulation from a to b.

Summary

Thus, the definite integral represents the area under the curve of a function on a specified interval, reflecting the accumulation of values of the function whether they contribute positively or negatively to the total area. This concept is fundamentally important in calculus and provides a powerful tool for understanding both geometric and physical phenomena.