Chapter 5: Applications of Integration

📚 Table of Contents

1. Area & Volume
2. Average Value
3. Cross Sections
4. Arc Length
5. Integration Applications

1. Area & Volume 📊

Understanding Area Applications

Area between curves represents accumulated difference. Think of it as:

• Net difference between functions
• Accumulated space
• Bounded regions
• Definite integral application

Finding Area

Between Two Curves

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

Process

1. Identify Functions

   • Determine upper/lower curves
   • Find intersection points
   • Check domain restrictions
   • Consider orientation

2. Set Up Integral

   • Choose appropriate bounds
   • Order functions correctly
   • Consider absolute value
   • Check for multiple regions

3. Evaluate

   • Use integration techniques
   • Verify result
   • Check reasonableness
   • Consider symmetry

Example Walkthrough

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

1. Graph curves:
   • y = x is linear
   • y = x² is parabola
   • x intersects at 0, 1
2. Compare functions:
   • x > x² on (0,1)
   • x is upper curve
3. Set up integral:
   • $A = \int_0^1 (x - x^2)dx$
4. Evaluate:
   • $= [\frac{x^2}{2} - \frac{x^3}{3}]_0^1$
   • $= (\frac{1}{2} - \frac{1}{3})$
   • $= \frac{1}{6}$

Volume Calculations

Understanding Methods

Think of volumes as:

• Accumulated cross-sections
• Rotated regions
• Sliced solids
• Three-dimensional accumulation

Disk Method

$V = \pi\int_a^b [f(x)]^2dx$

When to Use

• Rotating around x-axis
• Simple function squared
• Circular cross sections
• Solid with no hole

Washer Method

$V = \pi\int_a^b [R(x)^2 - r(x)^2]dx$

When to Use

• Rotating around axis
• Region between curves
• Hollow objects
• Nested cylinders

Shell Method

$V = 2\pi\int_a^b xf(x)dx$

When to Use

• Rotating around y-axis
• Complex functions
• Multiple regions
• Often simpler integration

Method Selection Guide

1. Choose Disk/Washer When:

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

2. Choose Shell When:

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

Example Walkthrough

Find volume when y = x² is rotated about y-axis from y = 0 to y = 4

1. Analyze problem:
   • Rotating around y-axis
   • Need x in terms of y
   • Shell method best
2. Set up:
   • x = ±√y
   • Use shell method
3. Integrate:
   • $V = 2\pi\int_0^4 \sqrt{y} \cdot y dy$
   • $= 2\pi[\frac{2y^{5/2}}{5}]_0^4$

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

2. Average Value

Understanding Average Value

The average value of a function over an interval. Think of it as:

• Average height of a curve
• Equally distributed value
• Definite integral application

Formula

$\bar{f} = \frac{1}{b-a}\int_a^b f(x)dx$

Process

1. Set Up Integral

   • Determine bounds
   • Check units

2. Evaluate

   • Use integration techniques
   • Divide by interval length
   • Verify units
   • Check reasonableness

Example Walkthrough

Find average value of f(x) = x² on [0,2]

1. Set up integral:
   • $\bar{f} = \frac{1}{2-0}\int_0^2 x^2 dx$
2. Evaluate:
   • $= \frac{1}{2}\int_0^2 x^2 dx$
   • $= \frac{1}{2}[\frac{x^3}{3}]_0^2$
   • $= \frac{1}{2}[\frac{8}{3} - 0]$
   • $= \frac{4}{3}$

3. Cross Sections

Understanding Cross Sections

The area of a cross-section of a solid. Think of it as:

• Slice of a solid
• Definite integral application

Formula

$A = \int_a^b f(x)dx$

Process

1. Identify Function

   • Function describing cross-section
   • Check domain restrictions
   • Consider orientation

2. Set Up Integral

   • Choose appropriate bounds
   • Check units

3. Evaluate

   • Use integration techniques
   • Verify result
   • Check reasonableness

Example Walkthrough

Find area of cross-section of solid with f(x) = x²

1. Set up integral:
   • $A = \int_0^2 x^2 dx$
2. Evaluate:
   • $= [\frac{x^3}{3}]_0^2$
   • $= \frac{8}{3}$

4. Arc Length 📐

Understanding Arc Length

The true distance along a curve. Think of it as:

• Path length
• Curve measurement
• Accumulated distance
• True curve size

Formula

$L = \int_a^b \sqrt{1 + [f'(x)]^2}dx$

Process

1. Find Derivative

   • Calculate f'(x)
   • Square the derivative
   • Add 1
   • Take square root

2. Set Up Integral

   • Determine bounds
   • Simplify if possible
   • Consider substitution
   • Check complexity

3. Evaluate

   • Choose technique
   • Watch for special cases
   • Verify units
   • Check reasonableness

Example Walkthrough

Find length of y = x² from x = 0 to x = 1

1. Find f'(x):
   • f'(x) = 2x
2. Set up integral:
   • $L = \int_0^1 \sqrt{1 + 4x^2}dx$
3. Evaluate:
   • Use substitution
   • u = 2x
   • Result involves ln

5. Integration Applications

Understanding Integration Applications

Integration has various applications in calculus. Think of it as:

• Solving problems
• Finding areas
• Calculating volumes
• Modeling real-world scenarios

Common Applications

1. Area Between Curves
   • $A = \int_a^b |f(x) - g(x)|dx$
2. Volume of Revolution
   • $V = \pi\int_a^b [f(x)]^2dx$
3. Average Value
   • $\bar{f} = \frac{1}{b-a}\int_a^b f(x)dx$
4. Cross Sections
   • $A = \int_a^b f(x)dx$

Example Walkthrough

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

1. Graph curves:
   • y = x is linear
   • y = x² is parabola
   • x intersects at 0, 1
2. Compare functions:
   • x > x² on (0,1)
   • x is upper curve
3. Set up integral:
   • $A = \int_0^1 (x - x^2)dx$
4. Evaluate:
   • $= [\frac{x^2}{2} - \frac{x^3}{3}]_0^1$
   • $= (\frac{1}{2} - \frac{1}{3})$
   • $= \frac{1}{6}$

📝 AP-Style Examples

Example 1: Area Between Curves

Find area between $y = x^2$ and $y = \sqrt{x}$ from x = 0 to x = 1

Solution:

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

Example 2: Volume of Revolution

Find volume when $y = \sin x$ is rotated about x-axis from x = 0 to x = π

Solution:

1. Choose method:
   • Rotating around x-axis
   • Use disk method
2. Set up integral:
   • $V = π\int_0^π [\sin x]^2dx$
3. Evaluate:
   • $= π\int_0^π \frac{1 - \cos(2x)}{2}dx$
   • $= π[\frac{x}{2} - \frac{\sin(2x)}{4}]_0^π$
   • $= \frac{π^2}{2}$

Example 3: Average Value

Find average value of $f(x) = \cos x$ on $[0,\frac{π}{2}]$

Solution:

1. Apply formula:
   • $\bar{f} = \frac{2}{π}\int_0^{π/2} \cos x dx$
2. Evaluate:
   • $= \frac{2}{π}[\sin x]_0^{π/2}$
   • $= \frac{2}{π}$

💡 Success Strategies

1. Setting Up Integrals

• Draw and label diagrams
• Identify correct bounds
• Choose appropriate method
• Verify integrand

2. Common Mistakes

• Wrong integration method
• Incorrect bounds
• Missing absolute value
• Function order confusion

3. Calculator Tips

• Graph to verify regions
• Check intersections
• Confirm reasonableness
• Use numerical integration

🔍 AP Exam Focus

Free Response Tips

1. Show all work:

   • Draw and label diagrams
   • Write correct integrals
   • Show evaluation steps
   • State final answer

2. Common Questions:

   • Area between curves
   • Volume of revolution
   • Average value
   • Cross sections

Multiple Choice Strategy

1. Analyze the Problem:

   • Read carefully
   • Draw quick sketch
   • Consider all methods
   • Look for patterns

2. Check Answer:

   • Reasonable magnitude
   • Correct sign
   • Appropriate units
   • Matches graph

📊 Quick Reference

Area Formulas

1. Between curves: $\int_a^b |f(x) - g(x)|dx$
2. With respect to y: $\int_c^d |h(y) - k(y)|dy$

Volume Formulas

1. Disk Method: $π\int_a^b [f(x)]^2dx$
2. Washer Method: $π\int_a^b [R(x)^2 - r(x)^2]dx$
3. Shell Method: $2π\int_a^b xf(x)dx$

Other Applications

1. Average Value: $\frac{1}{b-a}\int_a^b f(x)dx$
2. Arc Length: $\int_a^b \sqrt{1 + [f'(x)]^2}dx$

💡 Pro Tip: Always sketch the region first - it helps identify the correct integration method and bounds!