Definite Integrals
Learn to evaluate definite integrals using the Fundamental Theorem of Calculus, a cornerstone of HSC Advanced Mathematics.
The Fundamental Theorem of Calculus
A definite integral has upper and lower limits of integration. Unlike an indefinite integral, it produces a numerical value rather than a function. The Fundamental Theorem of Calculus states:
∫ab f(x) dx = F(b) − F(a)
where F(x) is any anti-derivative of f(x). We write this as [F(x)]ab.
Notice that the constant of integration C cancels out: F(b) + C − (F(a) + C) = F(b) − F(a). This is why we do not write +C for definite integrals.
Example: Evaluate ∫13 2x dx
Step 1: Find the anti-derivative: F(x) = x2
Step 2: Evaluate: [x2]13 = 32 − 12 = 9 − 1
Answer: 8
Evaluating Definite Integrals Step by Step
Find the anti-derivative F(x) of f(x).
Substitute the upper limit b into F(x) to get F(b).
Substitute the lower limit a into F(x) to get F(a).
Calculate F(b) − F(a) to obtain the value.
Important: Always subtract the value at the lower limit from the value at the upper limit: upper minus lower.
Properties of Definite Integrals
1. ∫aa f(x) dx = 0 — integral over zero width is zero
2. ∫ab f(x) dx = −∫ba f(x) dx — swapping limits changes the sign
3. ∫ab f(x) dx + ∫bc f(x) dx = ∫ac f(x) dx — intervals can be joined
4. ∫ab k · f(x) dx = k · ∫ab f(x) dx — constant multiple rule
Key Vocabulary
Definite Integral
An integral with upper and lower limits that evaluates to a specific number.
Limits of Integration
The values a (lower) and b (upper) that bound the region of integration.
Fundamental Theorem
The theorem connecting integration and differentiation: ∫ab f(x) dx = F(b) − F(a).
Evaluation Notation
[F(x)]ab means "F(x) evaluated from a to b", i.e. F(b) − F(a).
Worked Examples
Evaluate ∫02 (3x2 + 1) dx.
Step 1: Anti-derivative: F(x) = x3 + x
Step 2: F(2) = 8 + 2 = 10
Step 3: F(0) = 0 + 0 = 0
Answer: 10 − 0 = 10
Evaluate ∫1e (1/x) dx.
Step 1: Anti-derivative: F(x) = ln|x| = ln(x) for x > 0
Step 2: F(e) = ln(e) = 1
Step 3: F(1) = ln(1) = 0
Answer: 1 − 0 = 1
Evaluate ∫0π/2 cos(x) dx.
Step 1: Anti-derivative: F(x) = sin(x)
Step 2: F(π/2) = sin(π/2) = 1
Step 3: F(0) = sin(0) = 0
Answer: 1 − 0 = 1
Knowledge Check
Select the correct answer for each question. Click "Check Answer" to see if you are right.
Question 1
Evaluate ∫03 x2 dx.
Question 2
Evaluate ∫01 ex dx.
Question 3
What happens when you swap the limits of a definite integral?
Question 4
Evaluate ∫0π sin(x) dx.
Question 5
Evaluate ∫14 (2x + 1) dx.
Key Concepts Summary
- ● The Fundamental Theorem of Calculus: ∫ab f(x) dx = F(b) − F(a).
- ● Definite integrals produce a number, not a function. No +C is needed.
- ● Swapping the limits reverses the sign of the integral.
- ● Adjacent integrals can be combined by joining their intervals.
- ● Always evaluate carefully: upper limit value minus lower limit value.