Definite Integrals

A definite integral calculates the exact signed area between a function and the x-axis over a specific interval. Unlike indefinite integrals, the result is a number, not a function.

Notation and Definition

A definite integral is written with upper and lower limits:

Definite Integral Notation

$$\int_{a}^{b} f(x) \, dx$$
  • a = lower limit (starting point)
  • b = upper limit (ending point)
  • f(x) = the integrand

Evaluating Definite Integrals

The Fundamental Theorem of Calculus (Part 2) provides the method:

Evaluation Formula

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

Where F(x) is any antiderivative of f(x)

Steps to Evaluate

  1. Find the antiderivative F(x) of f(x)
  2. Evaluate F at the upper limit: F(b)
  3. Evaluate F at the lower limit: F(a)
  4. Subtract: F(b) - F(a)

Example

Evaluate: ∫₀² x² dx

Solution:

  • Antiderivative: F(x) = x³/3
  • F(2) = 8/3
  • F(0) = 0
  • Answer: 8/3 - 0 = 8/3

Properties of Definite Integrals

Order of Limits

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

Zero Width Interval

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

Additivity

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

Constant Multiple

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

Sum/Difference

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

Geometric Interpretation

The definite integral represents the signed area between the curve and the x-axis:

  • Area above the x-axis is positive
  • Area below the x-axis is negative

Finding Total Area (Not Signed)

To find the total area regardless of sign:

  1. Find where f(x) = 0 within [a, b]
  2. Split the integral at these points
  3. Take the absolute value of each part
  4. Add them together
$$\text{Total Area} = \int_{a}^{b} |f(x)| \, dx$$

Average Value of a Function

Average Value Formula

$$f_{avg} = \frac{1}{b-a} \int_{a}^{b} f(x) \, dx$$

The average value of f(x) on the interval [a, b]

Substitution with Definite Integrals

When using u-substitution with definite integrals, you have two options:

Option 1: Change the Limits

Convert the limits from x-values to u-values:

$$\int_{a}^{b} f(g(x)) \cdot g'(x) \, dx = \int_{g(a)}^{g(b)} f(u) \, du$$

Option 2: Substitute Back

  1. Perform substitution
  2. Find the antiderivative in terms of u
  3. Substitute back to get F(x)
  4. Evaluate at original limits a and b

More Examples

Example: Trigonometric Function

Evaluate: ∫₀^π sin(x) dx

Solution:

  • Antiderivative: -cos(x)
  • [-cos(x)]₀^π = -cos(π) - (-cos(0))
  • = -(-1) - (-1) = 1 + 1
  • Answer: 2

Example: Exponential Function

Evaluate: ∫₀¹ eˣ dx

Solution:

  • Antiderivative: eˣ
  • [eˣ]₀¹ = e¹ - e⁰ = e - 1
  • Answer: e - 1 ≈ 1.718
Indefinite Integrals Applications