📊 Chapter 5: Applications of Integration

AP Exam Weight: 20-30% | Multiple Choice: 8-12 questions | Free Response: Major focus in several questions

📚 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 = ∫ab|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 = ∫01(x − x2)dx

4. Evaluate:

• $= [\frac{x^2}{2} - \frac{x^3}{3}]_0^1$
• $= (\frac{1}{2} - \frac{1}{3})$
• $= \frac{1}{6}$