📐 Concept diagram

05-02 - The Definite Integral

Phase: 5 | Subject: 05-02 Prerequisites: 05-01-antiderivatives.md (indefinite integrals) Next subject: 05-03-the-fundamental-theorem-of-calculus.md

Learning Objectives

By the end of this subject, you will be able to:

1. Understand the definite integral as area under a curve
2. Use sigma notation for Riemann sums
3. Evaluate left, right, and midpoint Riemann sums
4. Understand the limit definition of the definite integral
5. Apply basic properties of definite integrals

Core Content

From Indefinite to Definite

Indefinite integral: ∫f(x)dx = F(x) + C (family of antiderivatives) Definite integral: ∫[a to b] f(x)dx = F(b) - F(a) (a NUMBER — the signed area)

Area Under a Curve

The definite integral ∫[a to b] f(x)dx represents the net signed area between the curve y = f(x), the x-axis, and the vertical lines x = a and x = b.

Signed area: Areas above the x-axis are positive, areas below are negative.

Example: ∫[0 to 2] x dx = [x²/2][0 to 2] = 4/2 - 0 = 2 This is the area of a triangle: (1/2) × base × height = (1/2) × 2 × 2 = 2. ✓

Riemann Sums

Approximate area using rectangles.

Left Riemann sum: Use function value at the LEFT endpoint of each subinterval. Right Riemann sum: Use function value at the RIGHT endpoint. Midpoint Riemann sum: Use function value at the MIDPOINT.

Setup

1. Partition [a, b] into n subintervals of equal width Δx = (b - a)/n
2. Sample point xᵢ* in each subinterval
3. Sum: Σ f(xᵢ*)·Δx

Example: Approximate ∫[0 to 2] x² dx with n = 4 left endpoints.

Δx = 2/4 = 0.5 Points: 0, 0.5, 1, 1.5 f(0) = 0, f(0.5) = 0.25, f(1) = 1, f(1.5) = 2.25 Sum = (0 + 0.25 + 1 + 2.25) × 0.5 = 3.5 × 0.5 = 1.75

True value: ∫[0 to 2] x² dx = [x³/3][0 to 2] = 8/3 ≈ 2.667 Left sum underestimates for increasing function.

Limit Definition

$∫[a to b] f(x)dx = $lim(n→∞)$ Σ(i=1 to n) f(xᵢ*)·Δx
$

As n → ∞, the approximation becomes exact.

Properties of Definite Integrals

1. ∫[a to a] f(x)dx = 0
2. ∫[a to b] f(x)dx = -∫[b to a] f(x)dx
3. ∫[a to b] [f(x) ± g(x)]dx = ∫[a to b] f(x)dx ± ∫[a to b] g(x)dx
4. ∫[a to b] k·f(x)dx = k·∫[a to b] f(x)dx
5. ∫[a to b] f(x)dx = ∫[a to c] f(x)dx + ∫[c to b] f(x)dx
6. If f(x) ≥ 0 on [a, b], then ∫[a to b] f(x)dx ≥ 0
7. If f(x) ≤ g(x) on [a, b], then ∫[a to b] f(x)dx ≤ ∫[a to b] g(x)dx

Key Terms

• 05 02 The Definite Integral
• Area Under a Curve
• Correct: A)
• Correct: B)
• Correct: C)
• Example 1: Geometric interpretation
• Example 2: Using properties
• Example 3: Definite integral evaluation
• From Indefinite to Definite
• Limit Definition
• Properties of Definite Integrals
• Riemann Sums

Worked Examples

Example 1: Geometric interpretation

Find ∫[-2 to 2] |x| dx.

Split at x = 0: ∫[-2 to 0] (-x)dx + ∫[0 to 2] x dx = [-x²/2][-2 to 0] + [x²/2][0 to 2] = (0 - (-2)) + (2 - 0) = 2 + 2 = 4

This is the area of two triangles: each has base 2, height 2. Total = 2 × (1/2 × 2 × 2) = 4. ✓

Example 2: Using properties

If ∫[1 to 3] f(x)dx = 5 and ∫[3 to 5] f(x)dx = 7, find ∫[1 to 5] f(x)dx.

∫[1 to 5] = ∫[1 to 3] + ∫[3 to 5] = 5 + 7 = 12

Example 3: Definite integral evaluation

∫[0 to 4] (x² − 2x)dx

= [x³/3 − x²][0 to 4] = (64/3 − 16) − (0 − 0) = 64/3 − 48/3 = 16/3

Practice Problems

Problem 1: Evaluate ∫[0 to 1] (3x² + 2x)dx

Problem 2: Evaluate ∫[1 to 4] (4/x)dx

Problem 3: Evaluate ∫[-1 to 1] x³ dx

Problem 4: Geometric: ∫[0 to π] sin(x)dx

Problem 5: If ∫[0 to 2] f(x)dx = 4 and ∫[2 to 5] f(x)dx = 9, find ∫[0 to 5] f(x)dx

Summary

Key takeaways:

• ∫[a to b] f(x)dx = F(b) - F(a) (net signed area)
• Riemann sums approximate area using rectangles
• Left/right/midpoint sums converge to the integral as n → ∞
• Properties: linearity, additivity, reversal

Pitfalls

• Confusing indefinite and definite integrals. An indefinite integral is a function (family of antiderivatives with +C). A definite integral is a number (signed area). Mixing up the two leads to leaving +C on a definite integral or forgetting it on an indefinite one.
• Ignoring that area below the x-axis is negative. The definite integral gives net signed area. If the curve dips below the x-axis, those regions subtract from the total. To find total geometric area, split the integral at x-intercepts and use absolute values.
• Using the wrong sample points in Riemann sums. Left Riemann sums use left endpoints, right sums use right endpoints, midpoint sums use midpoints. Each gives a different approximation — confirm which one is asked for.
• Reversing limits incorrectly. ∫[b to a] f(x)dx = -∫[a to b] f(x)dx. The negative sign is easy to forget, but swapping limits flips the sign of the entire integral.
• Assuming integrability without checking. Not every function is Riemann integrable. Functions must be bounded on [a,b] and have at most countably many discontinuities. Unbounded functions (e.g., 1/x on [0,1]) need improper integrals.

Quiz

Q1: ∫[1 to 3] 2x dx equals:

A) 4 B) 6 C) 8 D) 10

Q2: ∫[0 to 2] (x + 1)dx equals:

A) 2 B) 3 C) 4 D) 5

Q3: The integral ∫[a to b] f(x)dx represents:

A) The derivative of f(x) B) The net signed area under f(x) from a to b C) The average value of f(x) D) The length of the curve

Q4: If f(x) ≥ 0 on [a, b], then ∫[a to b] f(x)dx is:

A) Always positive B) Always negative C) Always zero D) Could be positive, negative, or zero

Q5: ∫[2 to 4] f(x)dx = 6 and ∫[4 to 7] f(x)dx = 9. Then ∫[2 to 7] f(x)dx = ?

A) 3 B) 15 C) 54 D) -15

Next Steps

Next up: 05-03-the-fundamental-theorem-of-calculus.md