Unit 6: Integration and Accumulation of Change

Master the art of integration and understand accumulation functions

← Back to Units

📋 Unit Overview

Unit 6 focuses on integration as accumulation and the fundamental theorem of calculus. You'll learn how to:

  • Understand integration as accumulation of change
  • Apply the Fundamental Theorem of Calculus
  • Use integration to find areas and volumes
  • Work with accumulation functions
  • Solve problems involving net change
💡 Key Insight: Integration is the reverse of differentiation and represents the accumulation of change over an interval. It's the mathematical tool for finding areas, volumes, and total change.

📈 Integration as Accumulation

What is Integration?

Integration represents the accumulation of change over an interval. If f(x) represents a rate of change, then the integral gives us the total change.

∫ab f(x) dx = F(b) - F(a)

where F'(x) = f(x)

🎯 Key Concept

Integration is the reverse of differentiation. If f(x) is the rate of change, then ∫f(x)dx gives the total change.

📝 Example 6.1: Accumulation of Change

If v(t) = 3t² represents velocity (rate of change of position), find the total distance traveled from t = 1 to t = 3.

Solution: ∫13 3t² dt = [t³]13 = 27 - 1 = 26 units

The Fundamental Theorem of Calculus

Part 1: Derivative of Integral
d/dx[∫ax f(t) dt] = f(x)

The derivative of an integral is the original function

Part 2: Evaluation of Integral
∫ab f(x) dx = F(b) - F(a)

where F'(x) = f(x)

📝 Example 6.2: FTC Application

Find d/dx[∫0x t² dt]

Solution: By FTC Part 1, d/dx[∫0x t² dt] = x²

🔄 Accumulation Functions

An accumulation function represents the accumulated change from a starting point to a variable endpoint:

A(x) = ∫ax f(t) dt

A(x) gives the accumulated change from a to x

🔑 Important Properties
  • A(a) = 0 (no accumulation at the starting point)
  • A'(x) = f(x) (rate of accumulation equals the integrand)
  • A(x) is continuous if f(x) is continuous
📝 Example 6.3: Accumulation Function

Let A(x) = ∫2x (3t - 1) dt. Find A(5) and A'(x).

Solution:

  • A(5) = ∫25 (3t - 1) dt = [3t²/2 - t]25 = (75/2 - 5) - (6 - 2) = 32.5 - 4 = 28.5
  • A'(x) = 3x - 1 (by FTC Part 1)

📐 Applications of Integration

Area Under a Curve

The definite integral gives the net area between the curve and the x-axis:

Area = ∫ab f(x) dx

Positive areas above x-axis, negative areas below

📝 Example 6.4: Finding Area

Find the area between y = x² and the x-axis from x = 0 to x = 2.

Solution: Area = ∫02 x² dx = [x³/3]02 = 8/3 square units

Net Change

Integration gives the net change in a quantity over time:

Net Change = ∫ab f'(x) dx = f(b) - f(a)

Total change equals final value minus initial value

📝 Example 6.5: Net Change

If the rate of change of a population is P'(t) = 1000e^(0.02t), find the net change in population from t = 0 to t = 10.

Solution: Net Change = ∫010 1000e^(0.02t) dt = [50000e^(0.02t)]010 = 50000(e^0.2 - 1) ≈ 11,070

🛠️ Integration Techniques

1. Power Rule
∫ xⁿ dx = x^(n+1)/(n+1) + C

for n ≠ -1

2. Substitution
∫ f(g(x))g'(x) dx = F(g(x)) + C

where F'(u) = f(u)

3. Integration by Parts
∫ u dv = uv - ∫ v du

LIATE rule for choosing u

4. Partial Fractions
For rational functions

Decompose into simpler fractions

🧮 Practice Problems

Problem 1

Evaluate ∫14 (2x + 3) dx

Solution:

∫14 (2x + 3) dx = [x² + 3x]14

= (16 + 12) - (1 + 3) = 28 - 4 = 24

Problem 2

Find d/dx[∫0x sin(t) dt]

Solution:

By the Fundamental Theorem of Calculus Part 1:

d/dx[∫0x sin(t) dt] = sin(x)

Problem 3

If A(x) = ∫1x t² dt, find A(3) and A'(x).

Solution:

A(3) = ∫13 t² dt = [t³/3]13 = 27/3 - 1/3 = 26/3

A'(x) = x² (by FTC Part 1)

💡 Exam Tips

🎯 Multiple Choice
  • Remember FTC Part 1: derivative of integral = integrand
  • Use substitution when you see composite functions
  • Check your antiderivative by differentiating
  • Pay attention to bounds and signs
✍️ Free Response
  • Show all steps in your integration
  • Use proper notation for definite integrals
  • Include units in applied problems
  • Check your answer makes sense
⚠️ Common Mistakes:
  • Forgetting the constant of integration in indefinite integrals
  • Confusing accumulation functions with regular functions
  • Not applying FTC correctly
  • Mixing up bounds in definite integrals

🚀 Ready to Test Your Knowledge?

Take our comprehensive Unit 6 practice test with 15 multiple choice questions and 3 free response problems!

📊
15 MC Questions

Progressive difficulty levels

✍️
3 Free Response

Show your work

⏱️
45 Min Timer

Real exam conditions

🎯 Start Practice Test Now

Get instant feedback and detailed solutions!

➡️ What's Next?

Ready for the next challenge?

In Unit 7, you'll learn about differential equations and their applications.

Go to Unit 7: Differential Equations →