The Definite Integral Definition
From Approximation to Exactness
What happens when you use infinitely many rectangles to approximate area? The approximation becomes exact. This is the central idea of the definite integral: it’s the limit of Riemann sums as the number of rectangles approaches infinity.
This definition is profound because it transforms a geometric problem (finding area) into an algebraic limit. It also extends the concept of “area” to regions where the function might be negative, giving us a powerful tool for computing accumulated quantities in physics, economics, and beyond.
The key insight: The integral symbol $\int$ is an elongated S for “Sum”: it represents the limit of infinitely many infinitely small contributions.
Prerequisite Map
Quick Reference
| Property | Value |
|---|---|
| Concept | Integration |
| Course | MATH161 |
| Section | Stewart 4.2 |
| Difficulty | Intermediate |
| Time | ~20 minutes |
Key Concepts
The Definition
If $f$ is a function defined on $[a, b]$, the definite integral of $f$ from $a$ to $b$ is:
$$\boxed{\int_a^b f(x)\,dx = \lim_{n \to \infty} \sum_{i=1}^{n} f(x_i^*) \Delta x}$$
provided this limit exists and gives the same value for all choices of sample points $x_i^*$ in $[x_{i-1}, x_i]$.
If this limit exists, we say $f$ is integrable on $[a, b]$.
Anatomy of Integral Notation
$$\int_a^b f(x)\,dx$$
| Symbol | Name | Meaning |
|---|---|---|
| $\int$ | Integral sign | Indicates a limit of sums |
| $a$ | Lower limit | Start of the interval |
| $b$ | Upper limit | End of the interval |
| $f(x)$ | Integrand | The function being integrated |
| $dx$ | Differential | Indicates variable of integration; represents “$\Delta x$ in the limit” |
Important: The variable $x$ is a “dummy variable.” The integral has the same value regardless of which letter we use:
$$\int_a^b f(x)\,dx = \int_a^b f(t)\,dt = \int_a^b f(u)\,du$$
When Does the Integral Exist?
Theorem: If $f$ is continuous on $[a, b]$, then $f$ is integrable on $[a, b]$.
More generally, $f$ is integrable if it has at most finitely many jump discontinuities on $[a, b]$.
Geometric Interpretation
When $f(x) \geq 0$ on $[a, b]$:
$$\int_a^b f(x)\,dx = \text{Area under } y = f(x) \text{ from } x = a \text{ to } x = b$$
y
| ___
| / \ ← y = f(x)
| / \
| / shaded\
| / region \
+---+----------+--→ x
a b
Integral = shaded areaWhen $f$ takes both positive and negative values:
$$\int_a^b f(x)\,dx = A_1 - A_2$$
where $A_1$ is the area above the $x$-axis and $A_2$ is the area below.
This is called the net area or signed area.
Using the Definition to Evaluate Integrals
For simple functions, we can evaluate $\int_a^b f(x)\,dx$ directly:
Step 1: Set up $\Delta x = \frac{b-a}{n}$ and $x_i = a + i\Delta x$
Step 2: Write $\sum_{i=1}^{n} f(x_i)\Delta x$
Step 3: Use summation formulas to simplify
Step 4: Take $\lim_{n \to \infty}$
Essential Summation Formulas
| Sum | Formula |
|---|---|
| $\sum_{i=1}^{n} 1$ | $n$ |
| $\sum_{i=1}^{n} i$ | $\frac{n(n+1)}{2}$ |
| $\sum_{i=1}^{n} i^2$ | $\frac{n(n+1)(2n+1)}{6}$ |
| $\sum_{i=1}^{n} i^3$ | $\left[\frac{n(n+1)}{2}\right]^2$ |
Practice Problems
For the integral $\int_0^4 (3x^2 + 1)\,dx$, identify:
- The integrand
- The lower limit of integration
- The upper limit of integration
- The variable of integration
Evaluate each integral by interpreting it as an area:
- $\int_0^5 3\,dx$
- $\int_0^4 x\,dx$
Evaluate $\int_0^3 (x - 2)\,dx$ by interpreting the integral as a net area. Sketch the region and identify areas above and below the $x$-axis.
Use the definition of the definite integral (as a limit of Riemann sums) to evaluate $\int_0^2 x^2\,dx$.
You may use: $\sum_{i=1}^{n} i^2 = \frac{n(n+1)(2n+1)}{6}$
Express the following limit as a definite integral on the given interval, then evaluate it using geometry or known integral values.
$$\lim_{n \to \infty} \sum_{i=1}^{n} \sqrt{4 - \left(\frac{2i}{n}\right)^2} \cdot \frac{2}{n}$$
Hint: What are $a$, $b$, and $f(x)$? What curve does $y = f(x)$ represent?
Common Misconceptions
the definite integral $\int_a^b f(x)\,dx$ measures the height of $f$ at a representative point.
This is the height-vs-slope error. The definite integral accumulates infinitely many infinitely thin rectangle areas; the result is a net signed area, not a single function value. For example, $\int_0^2 x^2\,dx = \tfrac{8}{3}$ reflects accumulated area across the interval, not the value of $f$ at any particular $x$. The function value $f(1) = 1$ and the integral $\tfrac{8}{3}$ are unrelated in general.
the integral of a product equals the product of the individual integrals.
This is the multiplicative-not-additive error. In general $\int_a^b f(x)g(x)\,dx \neq \left(\int_a^b f(x)\,dx\right)\left(\int_a^b g(x)\,dx\right)$. For example, $\int_0^1 x \cdot x\,dx = \int_0^1 x^2\,dx = \tfrac{1}{3}$, while $\left(\int_0^1 x\,dx\right)^2 = \left(\tfrac{1}{2}\right)^2 = \tfrac{1}{4}$. The two values differ; integrals distribute over sums, not products.
Mastery Checklist
Mental Model
The Integral as Infinite Subdivision: Imagine slicing a loaf of bread into thinner and thinner slices. Each slice has width $dx$ (infinitesimally small) and height $f(x)$ (the function value). The integral $\int_a^b f(x)\,dx$ is the total “volume” of all these infinitely thin slices: it’s what you get when you add up infinitely many infinitely small pieces. The notation literally tells this story: $\int$ (sum), $f(x)$ (height), $dx$ (width).
Connections
Looking back:
- Riemann Sums are finite approximations; the integral is their limit
- Limits give meaning to “as $n \to \infty$”
Looking ahead:
- Properties of Definite Integrals let us manipulate integrals algebraically
- The Fundamental Theorem of Calculus connects integrals to antiderivatives, eliminating the need for limit calculations
Real-world connections:
- Distance traveled = $\int_a^b v(t)\,dt$ (velocity integrated over time)
- Total mass = $\int_a^b \rho(x)\,dx$ (density integrated over length)
- Work done = $\int_a^b F(x)\,dx$ (force integrated over distance)
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|---|---|---|
| Riemann Sums | Skills Index | Properties of Integrals |
Last updated: 2026-01-22