Chapter Study Guide: The Definite Integral

Learning Objectives

• 1.2.1 State the definition of the definite integral.
• 1.2.2 Explain the terms integrand, limits of integration, and variable of integration.
• 1.2.3 Explain when a function is integrable.
• 1.2.4 Describe the relationship between the definite integral and net area.
• 1.2.5 Use geometry and the properties of definite integrals to evaluate them.
• 1.2.6 Calculate the average value of a function.

Key Terms & Glossary

• Definite Integral: A formalization of the area under a curve, evaluated between two specific limits, yielding a specific numeric value.
• Integrable Function: A function $f(x) for which the limit of its Riemann sum exists over a given interval [a, b]$.
• Integrand: The function $f(x)$ that is being integrated.
• Limits of Integration: The boundaries $a$ (lower) and $b$ (upper) over which the integral is evaluated.
• Variable of Integration: A dummy variable (e.g., $dx$) that specifies with respect to which variable the integration is being performed.

The "Big Idea"

The definite integral generalizes the concept of finding the area under a curve. While Riemann sums initially required functions to be continuous and non-negative, the definite integral lifts these restrictions to model real-world scenarios more robustly. Unlike an indefinite integral (which represents a family of functions), a definite integral results in a specific number representing the net area between the function and the $x$-axis.

[!NOTE] Notation History: The integration symbol $\int$ is an elongated "S", introduced by Gottfried Wilhelm Leibniz, representing "sigma" or summation.

Formula / Concept Box

Hierarchical Outline

• 1. The Definite Integral
  • 1.1. Core Definition
    • Generalization of the Riemann Sum
    • Transitioning from limits of sums ($\lim_{n \to \infty}$) to exact area
  • 1.2. Anatomy of the Notation
    • Integral Symbol ($\int$)
    • Limits of Integration ($a$ to $b$)
    • Integrand ($f(x)$)
    • Variable of integration ($dx$)
  • 1.3. Evaluation Strategies
    • Using Geometric Formulas (circles, triangles, rectangles)
    • Applying Definite Integral Properties

Visual Anchors

Definition-Example Pairs

• Term: Limits of Integration
  • Definition: The specific boundary values on the $x$-axis over which the area is calculated.
  • Real-World Example: Calculating the total water flowed into a tank from $t=2$ hours to $t=5 hours. The limits of integration are a=2$ and $b=5$.
• Term: Variable of Integration
  • Definition: A dummy variable showing the variable of respect for integration.
  • Real-World Example: $\int_0^1 f(t) dt$ and $\int_0^1 f(x) dx$ yield the exact same numerical result. It simply marks the "axis" of calculation.

Worked Examples

Example 1: Using Geometric Formulas to Calculate Definite Integrals

Problem: Use the formula for the area of a circle to evaluate the area represented by the integral of a semicircle function with radius 3, from $x = -3$ to $x = 3$.

Step-by-Step Breakdown:

1. Identify the Geometry: The function describes a semicircle centered at the origin with radius $r = 3$.
2. Determine the Area Formula: The total area of a full circle is $A = \pi r^2$.
3. Adapt for Semicircle: Since the integral bounds cover exactly the upper half of the circle, we need half the area: $A_{semi} = \frac{1}{2} \pi r^2$.
4. Substitute and Solve: $\text{Area} = \frac{1}{2} \pi (3)^2 = \frac{9\pi}{2}$
5. Conclusion: The definite integral evaluates to $\frac{9\pi}{2}$.

Example 2: Using the Properties of the Definite Integral

Problem: Express the definite integral $\int_a^b [3f(x) + 2g(x)] dx$ as the sum of separate definite integrals.

Step-by-Step Breakdown:

1. Apply Sum Rule: Split the integral at the addition sign. $\int_a^b 3f(x) dx + \int_a^b 2g(x) dx$
2. Apply Constant Multiple Rule: Pull the constants (3 and 2) outside their respective integrals. $3\int_a^b f(x) dx + 2\int_a^b g(x) dx$

Comparison Tables

Checkpoint Questions

1. How does a definite integral fundamentally differ from an indefinite integral in terms of its final output?
2. If $\int_1^5 f(x) dx = 10$, what is the value of $\int_5^1 f(x) dx$ and which property defines this?
3. Why is the variable of integration referred to as a "dummy variable"?
4. How can you evaluate a definite integral if you do not know the algebraic integration rules yet?