Definite Integrals and FTC

🎯 Integration Basics: Definite Integrals and the Fundamental Theorem of Calculus

The Big Picture: Collecting Treasures in a Magical Garden 🌸

Imagine you have a magical garden. Every second, the garden grows flowers at different speeds. Sometimes fast, sometimes slow. A definite integral is like counting ALL the flowers that grew between breakfast and lunch.

You’re not just looking at one moment. You’re adding up everything that happened over a stretch of time!

1. Definite Integrals: Adding Up All the Slices 🍕

What Is a Definite Integral?

Think of slicing a pizza. Each slice has a different amount of cheese. A definite integral adds up ALL the cheese from the first slice to the last slice.

The symbol looks like this:

$\int_{a}^{b} f(x) , dx$

• a = where you START (first slice)
• b = where you STOP (last slice)
• f(x) = how much “stuff” you have at each point
• dx = tiny little pieces you add together

Simple Example

If a car travels at 60 mph for 2 hours:

$\int_{0}^{2} 60 , dt = 60 \times 2 = 120 \text{ miles}$

You added up all the tiny bits of distance!

2. Riemann Sums: Building Blocks Before the Real Thing 🧱

The Idea

Before we had fancy integration, mathematicians used Riemann sums — basically, stacking rectangles under a curve to estimate the area.

Think of it like this: You want to know how much water is in a wavy pool. You could put rectangular blocks in the pool and count them!

Area ≈ Rectangle 1 + Rectangle 2 + Rectangle 3 + ...

How It Works

1. Divide the interval [a, b] into n small pieces
2. Pick a height for each rectangle (using the function value)
3. Multiply width × height for each rectangle
4. Add them all up!

$\sum_{i=1}^{n} f(x_i) \cdot \Delta x$

Types of Riemann Sums

Example: Estimate area under f(x) = x² from 0 to 2 using 4 rectangles

Each rectangle has width = 0.5

Left sum ≈ 0² + 0.5² + 1² + 1.5² × 0.5 = 1.75

The actual integral is 8/3 ≈ 2.67. More rectangles = better accuracy!

3. FTC Part 1: The Accumulation Function 📈

The Magic Discovery

Here’s something amazing: If you’re adding up stuff as you go, the RATE at which your total grows equals the original function!

FTC Part 1 says:

If $F(x) = \int_{a}^{x} f(t) , dt$, then $F’(x) = f(x)$

Story Time

Imagine a piggy bank. Money flows in at rate f(t).

• $F(x)$ = total money saved from time a to time x
• $F’(x)$ = how fast your savings grow RIGHT NOW
• That rate equals exactly how fast money is flowing in: f(x)!

Example

If $F(x) = \int_{0}^{x} t^2 , dt$

Then $F’(x) = x^2$

The derivative of the accumulation function gives you back the original function!

4. FTC Part 2: The Evaluation Theorem 🎉

The Shortcut Everyone Loves

This is the part that makes integration actually doable!

$\int_{a}^{b} f(x) , dx = F(b) - F(a)$

Where F is the antiderivative of f.

What This Means

Instead of adding infinitely many tiny rectangles, just:

1. Find the antiderivative F(x)
2. Plug in the top number (b)
3. Plug in the bottom number (a)
4. Subtract!

Example

Find $\int_{1}^{3} 2x , dx$

1. Antiderivative of 2x is $x^2$
2. Plug in 3: $3^2 = 9$
3. Plug in 1: $1^2 = 1$
4. Subtract: $9 - 1 = 8$

Answer: 8 ✨

5. Net Change Theorem: The Journey Tracker 🚗

The Big Idea

The integral of a rate of change gives you the NET change (total change from start to finish).

$\int_{a}^{b} F’(x) , dx = F(b) - F(a)$

Real Life Example

A car’s velocity tells you how fast it’s going. The integral of velocity gives you total displacement.

If velocity = 3t from t=0 to t=4:

$\int_{0}^{4} 3t , dt = \frac{3t^2}{2} \Big|_{0}^{4} = \frac{48}{2} - 0 = 24$

The car moved 24 units forward (net change in position).

Important!

• If the rate is sometimes negative (car going backward), the integral gives NET change
• Total distance traveled might be different from displacement!

6. Definite Integral Properties: The Rules of the Game 📜

These shortcuts make calculations easier:

Property 1: Flip the Limits, Flip the Sign

$\int_{a}^{b} f(x) , dx = -\int_{b}^{a} f(x) , dx$

Going backwards? Just negate it!

Property 2: Same Start and End = Zero

$\int_{a}^{a} f(x) , dx = 0$

No distance traveled = nothing to add up!

Property 3: Split It Up

$\int_{a}^{b} f(x) , dx = \int_{a}^{c} f(x) , dx + \int_{c}^{b} f(x) , dx$

Break a journey into parts. Total = sum of parts.

Property 4: Constants Come Out

$\int_{a}^{b} c \cdot f(x) , dx = c \cdot \int_{a}^{b} f(x) , dx$

Property 5: Add Functions Separately

$\int_{a}^{b} [f(x) + g(x)] , dx = \int_{a}^{b} f(x) , dx + \int_{a}^{b} g(x) , dx$

7. Trapezoidal Rule: Better Rectangles 📐

The Problem with Rectangles

Rectangles have flat tops. But curves aren’t flat! This causes errors.

The Solution: Trapezoids!

Instead of rectangles, use trapezoids (shapes with slanted tops) that hug the curve better.

graph TD
    A["Divide into n intervals"] --> B["Draw trapezoids under curve"]
    B --> C["Each trapezoid follows the slope"]
    C --> D["Add up all trapezoid areas"]
    D --> E["Much better approximation!"]

The Formula

$\int_{a}^{b} f(x) , dx \approx \frac{\Delta x}{2} [f(x_0) + 2f(x_1) + 2f(x_2) + … + 2f(x_{n-1}) + f(x_n)]$

Where $\Delta x = \frac{b-a}{n}$

Example

Approximate $\int_{0}^{2} x^2 , dx$ using 4 trapezoids

• $\Delta x = 0.5$
• Points: x = 0, 0.5, 1, 1.5, 2
• Values: 0, 0.25, 1, 2.25, 4

$\approx \frac{0.5}{2} [0 + 2(0.25) + 2(1) + 2(2.25) + 4]$ $= 0.25 [0 + 0.5 + 2 + 4.5 + 4] = 0.25 \times 11 = 2.75$

Actual answer: 8/3 ≈ 2.67. We’re close!

Summary: Your Integration Toolkit 🧰

You Did It! 🎊

You now understand how to:

✅ Add up infinitely many tiny pieces (definite integrals) ✅ Approximate with rectangles (Riemann sums) ✅ Connect derivatives and integrals (FTC) ✅ Track total changes (Net Change Theorem) ✅ Use properties to simplify ✅ Get better estimates with trapezoids

Integration is just sophisticated adding. And now you’re a pro at it!