Integration Formulas and the Net Change Theorem: Chapter Study Guide — Calculus II: Integral Calculus - Integration, Series, and Parametric Equations Study Notes

Learning Objectives

By the end of this study guide, you should be able to:

Apply the basic integration formulas to evaluate indefinite and definite integrals.
Explain the significance of the Net Change Theorem in understanding accumulated change.
Use the Net Change Theorem to solve applied problems, distinguishing between total distance and net displacement.
Apply the integrals of odd and even functions over symmetric intervals to simplify calculations.

Key Terms & Glossary

Net Change Theorem: A theorem stating that when a quantity changes, its final value equals its initial value plus the definite integral of its rate of change.
Definite Integral: An integral with specified upper and lower limits that evaluates to a number representing the net signed area under a curve.
Indefinite Integral: An integral without limits that represents a family of functions (antiderivatives) differing by a constant, $+ C.
Displacement: The net change in position from an initial point, taking direction into account (can be positive, negative, or zero).
Total Distance: The actual path length traveled, calculated by integrating the absolute value of the velocity function.

The "Big Idea"

[!IMPORTANT] The fundamental concept of this section is that integration is accumulation.

If you know the rate at which something is changing (like a car's velocity, or water flowing into a tank), taking the definite integral of that rate over a period of time gives you the total accumulated change of that quantity. The Net Change Theorem is just a practical restatement of the Fundamental Theorem of Calculus (Part 2), making it the ultimate tool for real-world physics and engineering problems.

Formula / Concept Box

Concept	Formula / Rule	Explanation
Net Change Theorem	F(b) = F(a) + \int_{a}^{b} F'(x) dx	Final Value = Initial Value + Accumulated Change
Net Displacement	\int_{t_1}^{t_2} v(t) dt	Integrates standard velocity to find position change.
Total Distance	\int_{t_1}^{t_2} \|v(t)\| dt	Integrates absolute velocity (speed) to find actual ground covered.
Symmetric Even Integral	\int_{-a}^{a} f(x) dx = 2 \int_{0}^{a} f(x) dx$	If $f(-x) = f(x), the left and right side areas are identical.
Symmetric Odd Integral	\int_{-a}^{a} f(x) dx = 0$	If $f(-x) = -f(x), the positive and negative areas cancel out perfectly.

Hierarchical Outline

1. Basic Integration Review
- Definite vs. Indefinite: Indefinite yields a family of functions; definite yields a specific value.
- The Power Rule: Using fundamental rules like \int x^n dx = \frac{x^{n+1}}{n+1} + C.
2. The Net Change Theorem
- Derivation: Derived directly from the Fundamental Theorem of Calculus.
- Formula variations: Can be written as \int_a^b F'(x)dx = F(b) - F(a) $or$ F(b) = F(a) + \int_a^b F'(x)dx$.
3. Applications: Motion and Displacement
- Displacement: Integrates $v(t). Automatically accounts for positive/negative direction.
- Total Distance: Integrates |v(t)|$. Treats all movement as positive.
4. Symmetric Intervals and Function Parity
- Even Functions: Symmetric across the y-axis. Double the half-interval area.
- Odd Functions: Symmetric about the origin. Areas cancel to equal zero.

Visual Anchors

Diagram 1: Function Parity Decision Matrix

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Diagram 2: Displacement vs Total Distance Graph

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Definition-Example Pairs

Term: Net Change Theorem
- Definition: The integration of a rate of change gives the total accumulated change over that time.
- Real-World Example: A tank has 100 gallons of water. Water flows in at a rate of 5 gal/min for 10 minutes. The new volume is $100 + \int_{0}^{10} 5 , dt = 150$ gallons.
Term: Total Distance Traveled
- Definition: The definite integral of the absolute value of velocity.
- Real-World Example: A runner runs 3 miles east, then 2 miles west. Their displacement is 1 mile east, but their total distance traveled is $|3| + |-2| = 5 miles.
Term: Odd Function Property
- Definition: If a function satisfies f(-x) = -f(x), its integral from -a $to$ a is exactly zero.
- Real-World Example: Integrating the function y = x^3 $from$ x = -5 $to$ x = 5. The negative area under the x-axis on the left perfectly cancels the positive area on the right, yielding 0.

Worked Examples

Example 1: Finding Displacement and Distance

Problem: A car moves due north at 40 mph between 2 p.m. and 4 p.m., then moves south at 30 mph between 4 p.m. and 5 p.m. What is the car's net displacement? What is its total distance traveled?

Step 1: Set up the intervals and velocity. Let positive velocity be North. From t=0 $to$ t=2 $hours,$ v(t) = 40 $. From$ t=2 $to$ t=3 $hours,$ v(t) = -30.

Step 2: Calculate Net Displacement. Displacement = \int_0^2 40 , dt + \int_2^3 (-30) , dtDisplacement = [40t]_0^2 + [-30t]_2^3Displacement = (80 - 0) + (-90 - (-60)) = 80 - 30 = 50 miles North$.

Step 3: Calculate Total Distance. $Distance = \int_0^2 |40| \, dt + \int_2^3 |-30| \, dt$ $Distance = 80 + \int_2^3 30 \, dt = 80 + 30 = 110 miles$ .

[!NOTE] Notice how absolute value forces the negative velocity area to become positive, capturing the true "tread on the tires" distance.

Example 2: Integrating a Power Function

Problem: Use the power rule to integrate $f(x) = x^3$ over the interval $[-2, 2].

Step 1: Notice the Symmetry. The interval is [-2, 2], which is symmetric around the y-axis.

Step 2: Check Function Parity. f(-x) = (-x)^3 = -x^3 = -f(x). The function is odd.

Step 3: Apply the Rule. By the symmetric integral rule for odd functions, \int_{-2}^{2} x^3 , dx = 0.

Checkpoint Questions

Recall: What is the key difference between an indefinite integral and a definite integral?
Application: If water is draining from a pool at a rate of R(t) liters/minute, write the integral that represents the net change in the pool's volume from t=5 $to$ t=15 minutes.
Analysis: You are given a velocity graph where the area above the t-axis is 15 and the area below the t-axis is 5. What is the net displacement? What is the total distance traveled?
Concept Check: Why is the integral of an even function from -a $to$ a $equal to $2 \times \int_0^a f(x) dx$ ?

▶Click to expand Checkpoint Answers

An indefinite integral represents a family of functions (has $+ C), while a definite integral evaluates to a specific number (or single function) bounded by specific limits of integration.
The net change in volume is given by \int_{5}^{15} R(t) , dt$.
Net displacement = $15 - 5 = 10$. Total distance = $15 + |-5| = 20.
Even functions are symmetric across the y-axis. The area from -a to 0 is an exact mirror image of the area from 0 to a. Thus, calculating one side and doubling it provides the total area.

Muddy Points & Cross-Refs

Confusing Indefinite vs Definite: Students often forget to add +C to indefinite integrals or mistakenly add +C to definite integrals. Tip: If you see boundaries on the \int sign, you want a number at the end, not a function with +C$.
Cross-Reference: For a deeper dive into evaluating the limits of integration, review The Fundamental Theorem of Calculus, Part 2. To master the antiderivatives required for the Net Change Theorem, revisit basic power rules and substitution techniques.