The Ultimate Fundamental Theorem of Calculus Cheat Sheet: Complete Guide

Introduction to the Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus (FTC) is one of the most powerful and elegant results in mathematics, establishing the crucial connection between differentiation and integration. This relationship reveals that these two operations are essentially inverses of each other, forming the cornerstone of calculus. The FTC is vital because it:

• Provides a practical method for evaluating definite integrals without using limits of Riemann sums
• Establishes the inverse relationship between derivatives and integrals
• Enables the creation of antiderivative formulas and integration techniques
• Forms the foundation for many applications in physics, engineering, and other sciences

Core Concepts: The Two Parts of the FTC

Part 1: Differentiation of an Integral

If $f$ is continuous on $[a,b]$ and we define $F(x) = \int_a^x f(t) , dt$, then $F'(x) = f(x)$ for all $x \in [a,b]$.

Interpretation:

• $F(x)$ represents the accumulated area under the curve from $a$ to $x$
• The rate of change of this accumulated area is given by the original function value $f(x)$

Part 2: Evaluation of a Definite Integral

If $f$ is continuous on $[a,b]$ and $F$ is any antiderivative of $f$ (i.e., $F'(x) = f(x)$), then: $$\int_a^b f(x) , dx = F(b) – F(a)$$

Interpretation:

• The definite integral equals the net change in the antiderivative
• Often written using the notation: $\int_a^b f(x) , dx = [F(x)]_a^b = F(b) – F(a)$

Step-by-Step Process for Applying the FTC

Evaluating Definite Integrals (Part 2)

1. Find an antiderivative: Determine $F(x)$ such that $F'(x) = f(x)$
2. Evaluate at endpoints: Calculate $F(b) – F(a)$
3. Substitute values: Replace any parameters with their specific values

Working with Functions Defined by Integrals (Part 1)

1. Identify the integral function: Recognize when a function is defined as $F(x) = \int_a^x f(t) , dt$
2. Apply the FTC directly: Conclude that $F'(x) = f(x)$
3. Extend using the chain rule: For functions like $G(x) = \int_a^{g(x)} f(t) , dt$, determine that $G'(x) = f(g(x)) \cdot g'(x)$

Key Techniques and Applications

Computing Areas Under Curves

Computing Average Values

The average value of a function $f$ on $[a,b]$ is: $$f_{avg} = \frac{1}{b-a} \int_a^b f(x) , dx$$

Net Change Theorem

The net change in a quantity equals the integral of its rate of change:

If $F'(x) = f(x)$, then the net change in $F$ from $x = a$ to $x = b$ is: $$F(b) – F(a) = \int_a^b f(x) , dx$$

Applications include:

• Distance from velocity
• Work from force
• Mass from density

Variable Limits of Integration

For functions defined by integrals with variable limits:

$$F(x) = \int_{g(x)}^{h(x)} f(t) , dt$$

The derivative is given by: $$F'(x) = f(h(x)) \cdot h'(x) – f(g(x)) \cdot g'(x)$$

Common Forms and Special Cases

Substitution and the FTC

When using u-substitution with definite integrals, you can either:

1. Change the limits of integration to correspond to the new variable
2. Substitute back to the original variable before evaluating

Integrals Involving Symmetry

Common Challenges and Solutions

Challenge 1: Complex Integrands

Solution: Break down complicated functions using algebraic manipulation, then integrate each part separately.

Challenge 2: Improper Integrals

Approach:

• For infinite limits: $\int_a^{\infty} f(x) , dx = \lim_{t \to \infty} \int_a^t f(x) , dx$
• For discontinuities: $\int_a^b f(x) , dx = \lim_{t \to c^-} \int_a^t f(x) , dx + \lim_{t \to c^+} \int_t^b f(x) , dx$ where $c$ is the point of discontinuity

Challenge 3: Functions Defined by Integrals

Solution: Apply Part 1 of the FTC directly, possibly combined with the chain rule.

Challenge 4: Finding Antiderivatives

Solution: Develop proficiency with integration techniques (substitution, parts, partial fractions, etc.)

Comparison of Integration Techniques

Best Practices and Practical Tips

• Check your work: Differentiate your antiderivative to verify it equals the original function
• Use symmetry: Recognize even/odd functions to simplify calculations
• Drawing diagrams: Sketch the function to visualize the area being calculated
• Units analysis: When solving applied problems, track units through the integration
• Split complex integrals: Break complicated integrals into simpler parts
• Consider bounds carefully: Pay special attention to the limits of integration, especially with substitution
• Use technology wisely: Calculators/software can verify results but understanding the process is crucial

Common Mistakes to Avoid

• Forgetting to add the constant of integration for indefinite integrals
• Applying the wrong integration formula
• Miscalculating the antiderivative
• Incorrectly applying limits of integration
• Neglecting to use the chain rule when appropriate
• Failing to check if an improper integral converges

Advanced Applications of the FTC

• Analysis of motion: Position, velocity, and acceleration relationships
• Economics: Consumer and producer surplus calculations
• Probability: Calculating probabilities using density functions
• Physics: Work, energy, and fluid pressure calculations
• Engineering: Moments and centers of mass determination

Resources for Further Learning

Textbooks

• “Calculus” by James Stewart
• “Calculus: Early Transcendentals” by Jon Rogawski
• “Thomas’ Calculus” by George B. Thomas Jr.

Online Resources

• Khan Academy’s Calculus Courses
• MIT OpenCourseWare (18.01 Single Variable Calculus)
• Paul’s Online Math Notes (https://tutorial.math.lamar.edu/)
• 3Blue1Brown’s Essence of Calculus video series

Practice Resources

• “The Humongous Book of Calculus Problems” by W. Michael Kelley
• AP Calculus past exams
• Schaum’s Outlines: Calculus

Interactive Tools

• Desmos for graphing and visualization
• GeoGebra for dynamic mathematics applications
• Wolfram Alpha for computation and analysis

Key Formulas Reference

Basic Antiderivatives

• $\int x^n , dx = \frac{x^{n+1}}{n+1} + C$ (for $n \neq -1$)
• $\int \frac{1}{x} , dx = \ln|x| + C$
• $\int e^x , dx = e^x + C$
• $\int \sin x , dx = -\cos x + C$
• $\int \cos x , dx = \sin x + C$
• $\int \sec^2 x , dx = \tan x + C$
• $\int \frac{1}{1+x^2} , dx = \arctan x + C$

Properties of Definite Integrals

• $\int_a^b f(x) , dx = -\int_b^a f(x) , dx$
• $\int_a^b [f(x) + g(x)] , dx = \int_a^b f(x) , dx + \int_a^b g(x) , dx$
• $\int_a^b c \cdot f(x) , dx = c \cdot \int_a^b f(x) , dx$ (where $c$ is constant)
• $\int_a^c f(x) , dx + \int_c^b f(x) , dx = \int_a^b f(x) , dx$