13. Definite Integrals: Definition and Properties

A definite integral can be understood as the limit of the sum of areas of many small rectangles. Its properties help you split, combine, and simplify integrals over complex intervals.

I pay close attention to the direction of the limits of integration: reversing the upper and lower bounds flips the sign of the definite integral.

In the previous lesson, we covered the fundamentals of integration—including basic integration rules and the method of substitution. Today, we delve deeper into the concept, formal definition, and key properties of definite integrals. This material is essential for understanding how definite integrals are applied in practice—and serves as a solid foundation for the upcoming lesson, “Definite Integrals and Applications: The Relationship Between Integration and Area.”

Definition of the Definite Integral

A definite integral represents the accumulated value of a function over a specified interval. It is conventionally denoted by

$$\int_a^b f(x)\,dx,$$

which expresses the definite integral of $f(x)$ over the closed interval $[a, b]$.

When learning the definite integral, focus on five core ideas: partitioning the interval, evaluating the function, approximating area, taking the limit, and recognizing fundamental properties.

Formal Definition

The rigorous definition of the definite integral relies on the Riemann sum. Given an interval $[a, b]$, divide it into $n$ subintervals, each of width $\Delta x_i = x_i - x_{i-1}$, where $x_0 = a$ and $x_n = b$. Choose a sample point $c_i$ in each subinterval $[x_{i-1}, x_i]$. The corresponding Riemann sum is:

$$S_n = \sum_{i=1}^{n} f(c_i)\,\Delta x_i.$$

As $n \to \infty$—that is, as the number of subintervals grows without bound and their widths shrink toward zero—the limit of the Riemann sums defines the definite integral:

$$\int_a^b f(x)\,dx = \lim_{n \to \infty} S_n.$$

Geometric Interpretation

Geometrically, the definite integral represents the net signed area between the curve $y = f(x)$ and the $x$-axis over $[a, b]$. Areas above the $x$-axis contribute positively; those below contribute negatively. As a result, the definite integral may yield a negative value—or even zero—if positive and negative regions exactly cancel.

Properties of Definite Integrals

Several important properties underpin efficient computation and theoretical reasoning with definite integrals:

After reading “Definite Integrals and Applications: Definition and Properties,” take one minute to reflect: Are the key concepts clearly distinguished? Can you reproduce the solution steps? Can you restate the conclusions in your own words?

1. Linearity:
   If $c$ is a constant and $f(x), g(x)$ are integrable functions, then

   $$\int_a^b \big[ c f(x) + g(x) \big]\,dx = c \int_a^b f(x)\,dx + \int_a^b g(x)\,dx.$$

2. Additivity over Intervals:
   If $a < b < c$, then

$$\int_a^c f(x)\,dx = \int_a^b f(x)\,dx + \int_b^c f(x)\,dx.$$

Reversal of Limits:
Swapping the upper and lower limits introduces a minus sign:

$$\int_a^b f(x)\,dx = -\int_b^a f(x)\,dx.$$

Order (Inequality) Property:
If $m \leq f(x) \leq M$ for all $x \in [c, d]$, then

$$\int_c^d m\,dx \leq \int_c^d f(x)\,dx \leq \int_c^d M\,dx.$$

Equivalently, $m(d - c) \leq \int_c^d f(x)\,dx \leq M(d - c)$.

These properties allow us to manipulate and simplify definite integrals effectively.

Application Example: Computing a Definite Integral

Let’s now apply the definition and properties of definite integrals to compute a concrete example.

Example: Compute $\int_1^3 x^2\,dx$

We evaluate the definite integral of $f(x) = x^2$ over $[1, 3]$:

$$\int_1^3 x^2\,dx.$$

Step-by-Step Computation

1. Find an Antiderivative:
   An antiderivative $F(x)$ of $f(x) = x^2$ is

   $$F(x) = \frac{x^3}{3} + C.$$

2. Apply the Fundamental Theorem of Calculus:
   Using the Newton–Leibniz formula:

   $$\int_1^3 x^2\,dx = F(3) - F(1) = \left(\frac{3^3}{3}\right) - \left(\frac{1^3}{3}\right) = 9 - \frac{1}{3} = \frac{26}{3}.$$

3. Result:
   Therefore, $\int_1^3 x^2\,dx = \frac{26}{3}$.

Code Implementation

We can verify this result using Python and SymPy:

import sympy as sp

x = sp.symbols('x')
f = x**2
integral_value = sp.integrate(f, (x, 1, 3))
integral_value

Running this code returns $\frac{26}{3}$, confirming our manual calculation.

When reviewing “Definite Integrals and Applications: Definition and Properties,” place key concepts, procedural steps, and final results on the same page for efficient recall.

When practicing “Definite Integrals and Applications: Definition and Properties,” write the input conditions, computational actions, and observable outputs together—making future review straightforward.

Conclusion

This article introduced the definition and essential properties of definite integrals, illustrated with a concrete computational example. These foundations prepare you for the next lesson: “Definite Integrals and Applications: The Relationship Between Integration and Area.” In subsequent tutorials, we will explore how definite integrals quantify geometric area—and how these ideas power core techniques in artificial intelligence and machine learning. Through this series, you’ll develop both conceptual depth and practical fluency in applying calculus to real-world AI problems.