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The Definite Integral Definition

Reference: Stewart 4.2 • Chapter: 4 • Section: 2

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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 area

When $f$ 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

Level 1 Reading Integral Notation

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

Thought Process

The integral notation $\int_a^b f(x)\,dx$ has specific parts. The function between $\int$ and $dx$ is the integrand. The subscript and superscript on $\int$ are the limits. The variable in the differential tells you the integration variable.

Show Answer

(a) Integrand: $3x^2 + 1$

(b) Lower limit: $0$

(c) Upper limit: $4$

(d) Variable of integration: $x$

Level 2 Evaluating by Geometry

Evaluate each integral by interpreting it as an area:

$\int_0^5 3\,dx$
$\int_0^4 x\,dx$

Thought Process

For (a), $y = 3$ is a horizontal line, so the region is a rectangle. For (b), $y = x$ is a line through the origin, so the region under it from 0 to 4 is a triangle.

Show Answer

(a) The region under $y = 3$ from $x = 0$ to $x = 5$ is a rectangle:

$$\int_0^5 3\,dx = 3 \times 5 = 15$$

(b) The region under $y = x$ from $x = 0$ to $x = 4$ is a right triangle with base 4 and height 4:

$$\int_0^4 x\,dx = \frac{1}{2} \times 4 \times 4 = 8$$

Level 3 Net Area Calculation

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.

Thought Process

First, find where $y = x - 2$ crosses the $x$-axis (set $x - 2 = 0$). Then identify the regions above and below the axis. Calculate each area geometrically (they're triangles), and subtract.

Show Answer

The line $y = x - 2$ crosses the $x$-axis when $x = 2$.

    y
    |
  1 +              /
    |            /
  0 +----+----/----+→ x
    |   0    2    3
 -1 +       /
    |     /
 -2 +   /

Below the axis ($0 \leq x \leq 2$): Triangle with base 2 and height 2 $$A_2 = \frac{1}{2}(2)(2) = 2$$

Above the axis ($2 \leq x \leq 3$): Triangle with base 1 and height 1 $$A_1 = \frac{1}{2}(1)(1) = \frac{1}{2}$$

Net area: $$\int_0^3 (x-2)\,dx = A_1 - A_2 = \frac{1}{2} - 2 = -\frac{3}{2}$$

The negative result indicates more area lies below the $x$-axis than above.

Level 4 Evaluating Using the Definition

Use the definition of the definite integral (as a limit of Riemann sums) to evaluate $\int_0^2 x^2\,dx$.

You may use: $\sum_{i=1}^{n} i^2 = \frac{n(n+1)(2n+1)}{6}$

Thought Process

Set up the right Riemann sum with $\Delta x = \frac{2}{n}$ and $x_i = \frac{2i}{n}$. Then:

Write $f(x_i) = \left(\frac{2i}{n}\right)^2 = \frac{4i^2}{n^2}$
Form the sum $\sum_{i=1}^{n} \frac{4i^2}{n^2} \cdot \frac{2}{n}$
Factor out constants and apply the summation formula
Simplify and take the limit as $n \to \infty$

Show Answer

With $a = 0$, $b = 2$:

$\Delta x = \frac{2-0}{n} = \frac{2}{n}$
$x_i = 0 + i \cdot \frac{2}{n} = \frac{2i}{n}$

$$\int_0^2 x^2\,dx = \lim_{n \to \infty} \sum_{i=1}^{n} f(x_i)\Delta x$$

$$= \lim_{n \to \infty} \sum_{i=1}^{n} \left(\frac{2i}{n}\right)^2 \cdot \frac{2}{n}$$

$$= \lim_{n \to \infty} \sum_{i=1}^{n} \frac{4i^2}{n^2} \cdot \frac{2}{n}$$

$$= \lim_{n \to \infty} \frac{8}{n^3} \sum_{i=1}^{n} i^2$$

$$= \lim_{n \to \infty} \frac{8}{n^3} \cdot \frac{n(n+1)(2n+1)}{6}$$

$$= \lim_{n \to \infty} \frac{8n(n+1)(2n+1)}{6n^3}$$

$$= \lim_{n \to \infty} \frac{4(n+1)(2n+1)}{3n^2}$$

$$= \lim_{n \to \infty} \frac{4(2n^2 + 3n + 1)}{3n^2}$$

$$= \lim_{n \to \infty} \frac{8n^2 + 12n + 4}{3n^2}$$

$$= \lim_{n \to \infty} \left(\frac{8}{3} + \frac{12}{3n} + \frac{4}{3n^2}\right)$$

$$= \frac{8}{3}$$

Level 5 Expressing a Limit as an Integral

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?

Thought Process

Compare the limit to the definition $\int_a^b f(x)\,dx = \lim_{n \to \infty} \sum_{i=1}^{n} f(x_i)\Delta x$.

We have $\Delta x = \frac{2}{n}$, so $b - a = 2$. Since $x_i = \frac{2i}{n}$ and ranges from $\frac{2}{n}$ to $2$ as $i$ goes from $1$ to $n$, this suggests $a = 0$, $b = 2$.

The function being evaluated is $f(x) = \sqrt{4 - x^2}$. What is the graph of $y = \sqrt{4 - x^2}$?

Show Answer

Identifying the components:

$\Delta x = \frac{2}{n}$, so the interval has length $2$
$x_i = \frac{2i}{n}$ with $x_n = 2$, so $b = 2$ and $a = 0$
$f(x_i) = \sqrt{4 - x_i^2}$, so $f(x) = \sqrt{4 - x^2}$

The limit equals:

$$\int_0^2 \sqrt{4 - x^2}\,dx$$

Geometric interpretation: The equation $y = \sqrt{4 - x^2}$ is equivalent to $x^2 + y^2 = 4$ with $y \geq 0$. This is the upper semicircle of radius 2 centered at the origin.

The integral from $0$ to $2$ represents a quarter circle of radius $2$:

    y
  2 +     _____
    |    /     |
    |   |      |
    | (quarter |
    |  circle) |
    |          |
  0 +----------+→ x
    0          2

$$\int_0^2 \sqrt{4-x^2}\,dx = \frac{1}{4}\pi(2)^2 = \pi$$

Mastery Checklist

[ ] I can state the definition of the definite integral as a limit of Riemann sums
[ ] I can identify the parts of integral notation (integrand, limits, differential)
[ ] I understand the "dummy variable" concept
[ ] I can evaluate simple integrals geometrically (rectangles, triangles, circles)
[ ] I can interpret integrals as net (signed) area when $f$ takes negative values
[ ] I can set up and evaluate an integral using the definition with summation formulas
[ ] I can express a limit of sums as a definite integral

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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Last updated: 2026-01-22