Lesson 4: Definite Integrals and Word Problem

👨🏽‍🏫 Introduction

Imagine you’re calculating the exact area of a garden bed 🌱, the total distance a car has traveled 🚗, or the volume of water that has flowed from a tap 🚿. All these require a powerful tool — definite integration.

In this lesson, you’ll learn how to evaluate definite integrals to find the exact value of a quantity — whether it’s area under a curve, displacement, or total accumulation.

• 📐 Understand what a definite integral means
• 🧮 Learn how to calculate definite integrals
• 🔍 Solve real-life problems involving area, distance, and quantities

🧠 Key Concepts

📌 1. What is a Definite Integral?

• A definite integral calculates the exact value of an area or quantity between two limits.
• It has a lower and upper limit:
  
• This represents the area under the curve of
  
  from
  
  to
  .

📌 2. Fundamental Theorem of Calculus

To evaluate a definite integral:

Where

is the antiderivative (integral) of
.

📌 3. Real-World Uses

• 📏 Area under curves (geometry, land design)
• 🚗 Displacement from velocity functions
• 📊 Accumulated values like cost, population growth, etc.

🔍 Worked Examples

1. 1. Example 1 – Area Under a Curve:
      Evaluate
      .
      Step 1: Integrate
      →
      
      Step 2: Apply limits:
      

1. 1. Example 2 – Total Distance Traveled:
      A vehicle moves with velocity
      
      from
      
      to
      .
      Find the total distance covered.
      

1. Example 3 – Cost of Production:
   The marginal cost of producing a product is
   .
   Find the total cost of producing 5 units (assuming zero fixed cost).
   

✍🏽 Practice Exercises

🅰️ Section A – Evaluate Definite Integrals

🅱️ Section B – Real-Life Applications

1. The velocity of a moving object is
   .
   Find the distance traveled from
   
   to
   .
2. The marginal cost of a product is
   .
   Find the total cost of producing 3 units.

✅ Answers and Explanations

• A1:
  
• A2:
  
• B1:
  
• B2:
  

📌 Recap

• 🔢 Definite integrals compute exact quantities over an interval
• 🧮 Use
  
  to evaluate using antiderivatives
• 📊 Great for solving real-world problems involving area, cost, and distance

💭 Reflection Prompt

Have you ever tracked something over time — like how much fuel you’ve used or how many steps you’ve walked each day? Can you imagine calculating the total amount between two points? That’s the power of definite integration. ⏱️📈🚶‍♂️