Chapter 4: Integration

AP Exam Weight: 25-35% | Multiple Choice: 10-14 questions | Free Response: Major focus in several questions

📚 Table of Contents

1. Antiderivatives
2. Definite Integrals
3. Integration Techniques
4. Applications
5. Fundamental Theorem

1. Antiderivatives ��

Understanding Antiderivatives

The reverse process of differentiation. Think of it as:

• Finding the original function from its derivative
• Recovering distance from velocity
• Working backwards through changes
• Finding a family of related functions

Basic Rules

Power Rule Integration

$\int x^n dx = \frac{x^{n+1}}{n+1} + C$ (n ≠ -1)

Process

1. Add 1 to exponent
2. Divide by new exponent
3. Add constant of integration
4. Check by differentiating

Example Walkthrough

Find $\int 3x^2 dx$

1. Identify parts:
   • Coefficient: 3
   • Base: x
   • Exponent: 2
2. Apply rule:
   • Add 1 to exponent: x³
   • Divide by new exponent: $\frac{x^3}{3}$
   • Multiply by coefficient: $x^3$
3. Add C:
   • Final answer: $x^3 + C$
4. Verify:
   • Differentiate: $\frac{d}{dx}(x^3 + C) = 3x^2$

Common Antiderivatives

Exponential Functions

• $\int e^x dx = e^x + C$
• $\int a^x dx = \frac{a^x}{\ln a} + C$

Trigonometric Functions

• $\int \sin x dx = -\cos x + C$
• $\int \cos x dx = \sin x + C$
• $\int \sec^2 x dx = \tan x + C$
• $\int \csc^2 x dx = -\cot x + C$

Special Cases

• $\int \frac{1}{x} dx = \ln|x| + C$
• $\int \frac{1}{\sqrt{1-x^2}} dx = \arcsin x + C$
• $\int \frac{1}{1+x^2} dx = \arctan x + C$

2. Definite Integrals 🎯

Understanding Definite Integrals

The definite integral represents total accumulation. Think of it as:

• Area under a curve
• Total distance traveled
• Accumulated change
• Net accumulation over interval

Definition

$\int_a^b f(x)dx = \lim_{n \to \infty} \sum_{i=1}^n f(x_i^*)\Delta x$

Intuitive Understanding

• Sum of many tiny rectangles
• Better approximation as n increases
• Like slicing area into thin strips
• Limit of Riemann sums

Process for Riemann Sums

1. Divide interval into n subintervals
2. Choose sample points
3. Form sum of rectangles
4. Take limit as n → ∞

Example Walkthrough

Approximate $\int_0^2 x^2 dx$ with n = 4

1. Set up:
   • Δx = (2-0)/4 = 0.5
   • Points: 0, 0.5, 1, 1.5, 2
2. Calculate heights:
   • f(0) = 0
   • f(0.5) = 0.25
   • f(1) = 1
   • f(1.5) = 2.25
   • f(2) = 4
3. Form sum:
   • Σf(x)Δx = (0 + 0.25 + 1 + 2.25)·0.5
4. Compare to actual:
   • Approximation ≈ 2.67
   • Actual = 8/3 ≈ 2.67

Properties

Basic Properties

1. Reversal of Limits:

   • $\int_a^b f(x)dx = -\int_b^a f(x)dx$
   • Like walking backwards
   • Changes sign
   • Same magnitude

2. Additivity:

   • $\int_a^b [f(x) \pm g(x)]dx = \int_a^b f(x)dx \pm \int_a^b g(x)dx$
   • Integrals distribute
   • Like adding areas
   • Preserves operations

3. Constant Multiple:

   • $\int_a^b cf(x)dx = c\int_a^b f(x)dx$
   • Constants come out
   • Scaling property
   • Like area scaling

Advanced Properties

1. Split Interval:

   • $\int_a^b f(x)dx = \int_a^c f(x)dx + \int_c^b f(x)dx$
   • Any point c between a and b
   • Additive property
   • Useful for complex problems

2. Comparison:

   • If f(x) ≥ g(x), then $\int_a^b f(x)dx ≥ \int_a^b g(x)dx$
   • Area comparison
   • Inequality preservation
   • Important for estimation

Common Applications

1. Area:

   • Between curve and x-axis
   • Between two curves
   • Absolute value for total area

2. Distance:

   • From velocity
   • Total vs. net
   • Direction matters

3. Average Value:

   • $f_{avg} = \frac{1}{b-a}\int_a^b f(x)dx$
   • Mean value theorem
   • Representative value
   • Like center of mass

3. Integration Techniques 🔄

U-Substitution

Understanding the Method

Think of it as:

• Chain rule in reverse
• Simplifying complex integrals
• Pattern recognition
• Strategic substitution

Process

1. Identify Parts:

   • Look for composite function
   • Find derivative pattern
   • Choose u carefully
   • Consider du

2. Set Up:

   • Write u = inner function
   • Find du/dx
   • Solve for dx
   • Rewrite integral

3. Integrate:

   • Work in terms of u
   • Use basic antiderivatives
   • Keep track of constants
   • Remember +C

4. Back Substitute:

   • Replace u with original
   • Check answer
   • Verify domain
   • Simplify if needed

Example Walkthrough

Evaluate $\int x\cos(x^2)dx$

1. Choose u:
   • Let u = x²
   • du = 2x dx
   • x dx = du/2
2. Rewrite:
   • $\int \cos(u)\frac{du}{2}$
3. Integrate:
   • $\frac{1}{2}\sin(u) + C$
4. Substitute back:
   • $\frac{1}{2}\sin(x^2) + C$

Integration by Parts

Formula Understanding

$\int u dv = uv - \int v du$

Think of it as:

• Product rule backwards
• Strategic choice of parts
• Trading complexity
• Systematic approach

LIATE Order

Choose u from left:

• Logarithmic
• Inverse trig
• Algebraic
• Trigonometric
• Exponential

Process

1. Choose Parts:

   • Select u using LIATE
   • Find dv
   • Calculate v
   • Set up formula

2. New Integral:

   • Often simpler
   • Sometimes same form
   • May need iteration
   • Watch for patterns

4. Applications 📊

Understanding Applications

Integration solves real-world accumulation problems. Think of it as:

• Finding total quantities from rates
• Calculating areas and volumes
• Determining accumulated change
• Measuring physical quantities

Area Calculations

Between Curves

$A = \int_a^b [f(x) - g(x)]dx$

Process

1. Find Intersection Points:

   • Solve f(x) = g(x)
   • These are bounds
   • Check crossings
   • Verify domain

2. Set Up Integral:

   • Top minus bottom
   • Check orientation
   • Consider absolute value
   • Verify bounds

3. Evaluate:

   • Use appropriate technique
   • Check units
   • Verify reasonableness
   • Consider symmetry

Example Walkthrough

Find area between y = x² and y = x from x = 0 to x = 1

1. Identify curves:
   • Upper: f(x) = x
   • Lower: g(x) = x²
2. Set up integral:
   • $A = \int_0^1 (x - x^2)dx$
3. Evaluate:
   • $= [\frac{x^2}{2} - \frac{x^3}{3}]_0^1$
   • $= (\frac{1}{2} - \frac{1}{3}) - (0 - 0)$
   • $= \frac{1}{6}$ square units

Volume Calculations

Understanding Methods

1. Disk Method:

   • $V = \pi\int_a^b [f(x)]^2dx$
   • Rotating around x-axis
   • Circular cross sections
   • Like stacking circles

2. Washer Method:

   • $V = \pi\int_a^b [R(x)^2 - r(x)^2]dx$
   • Nested cylinders
   • Hollow shapes
   • Difference of disks

3. Shell Method:

   • $V = 2\pi\int_a^b xf(x)dx$
   • Cylindrical shells
   • Often easier
   • Alternative approach

Method Selection

1. Choose Disk/Washer When:

   • Rotating around horizontal line
   • Simple function squared
   • Clear outer/inner functions
   • Straightforward bounds

2. Choose Shell When:

   • Rotating around vertical line
   • Complex functions
   • Multiple regions
   • Easier integration

Common Mistakes

1. Setup Errors:

   • Wrong method choice
   • Incorrect radius
   • Wrong axis of rotation
   • Bound confusion

2. Calculation Errors:

   • Forgetting π
   • Square vs. squared function
   • Wrong substitution
   • Integration mistakes

Other Applications

Work and Pressure

• $W = \int_a^b F(x)dx$
• Force times distance
• Variable force
• Accumulation of effort

Average Value

• $f_{avg} = \frac{1}{b-a}\int_a^b f(x)dx$
• Mean value theorem
• Representative value
• Typical behavior

5. Fundamental Theorem of Calculus 📐

Understanding FTC

The bridge between derivatives and integrals. Think of it as:

• Undoing differentiation
• Connecting rates to totals
• Simplifying calculations
• Unifying calculus concepts

Part 1 (FTC1)

Statement

If F'(x) = f(x), then: $\frac{d}{dx}\int_a^x f(t)dt = f(x)$

Interpretation

• Derivative undoes integration
• Rate of accumulation
• Instantaneous change
• Local behavior

Applications

1. Finding Derivatives:

   • Of integral functions
   • Variable upper limit
   • Chain rule needed
   • Check conditions

2. Theoretical:

   • Proves relationships
   • Justifies shortcuts
   • Connects concepts
   • Foundation for calculus

Part 2 (FTC2)

Statement

$\int_a^b f(x)dx = F(b) - F(a)$

Understanding

• Evaluate at endpoints
• Net change
• Total accumulation
• Difference of states

Process

1. Find Antiderivative:

   • Look for patterns
   • Use basic rules
   • Consider techniques
   • Verify derivative

2. Evaluate:

   • Plug in bounds
   • Subtract results
   • Check signs
   • Verify units

Example Walkthrough

Evaluate $\int_0^2 x^3dx$

1. Find F(x):
   • $F(x) = \frac{x^4}{4}$
2. Evaluate bounds:
   • F(2) = 16/4 = 4
   • F(0) = 0
3. Subtract:
   • 4 - 0 = 4

Common Mistakes

1. Conceptual:

   • Confusing parts 1 and 2
   • Wrong variable use
   • Missing conditions
   • Integration errors

2. Computational:

   • Wrong antiderivative
   • Bound substitution
   • Sign errors
   • Constant confusion

Practice Strategies

1. Understanding:

   • Draw diagrams
   • Use analogies
   • Connect concepts
   • Verify results

2. Problem Solving:

   • Identify theorem needed
   • Check conditions
   • Show clear work
   • Verify answer

📝 AP-Style Examples

Example 1: FTC Application

Find $\frac{d}{dx}\int_2^x t^3dt$

Solution:

1. Use FTC1: Result is x³
2. No need to integrate
3. Upper limit becomes variable
4. Lower limit doesn't affect derivative

Example 2: Area Calculation

Find area between y = x² and y = √x from x = 0 to x = 1

Solution:

1. Compare functions:
   • √x > x² on (0,1)
2. Set up integral:
   • $\int_0^1 (\sqrt{x} - x^2)dx$
3. Integrate:
   • $[\frac{2x^{3/2}}{3} - \frac{x^3}{3}]_0^1$
4. Evaluate:
   • $\frac{2}{3} - \frac{1}{3} = \frac{1}{3}$

💡 Success Strategies

1. Integration Method Selection

• Try simplest method first
• Look for patterns
• Consider alternatives
• Check answer reasonableness

2. Common Mistakes

• Wrong technique choice
• Missing +C
• Bound errors
• Sign mistakes

3. Calculator Tips

• Check with numerical integration
• Graph to verify area
• Use for complex calculations
• Verify endpoints

🔍 AP Exam Focus

Free Response Tips

1. Show work:

   • Method selection
   • Step-by-step process
   • Final evaluation

2. Common Questions:

   • Area/Volume
   • FTC applications
   • Average value
   • Accumulation

Multiple Choice Strategy

1. Consider:

   • Integration techniques
   • Properties of integrals
   • FTC applications

2. Check:

   • Units
   • Sign
   • Reasonableness

💡 Pro Tip: Integration is the inverse of differentiation. Always check your answer by differentiating!