Lesson 3: The Definite Integral and FTC Part 1
Estimated time: 35-45 minutes
Learning Objectives
By the end of this lesson, you will be able to:
- Define the definite integral as the limit of Riemann sums
- Interpret the definite integral as signed area
- Apply basic properties of definite integrals
- State and apply the Fundamental Theorem of Calculus Part 1 (Evaluation Theorem)
Definition of the Definite Integral
The Definite Integral
If f is a function defined on [a, b], the definite integral of f from a to b is:
∫ab f(x) dx = limn→∞ ∑i=1n f(xi*) Δx
provided this limit exists (in which case f is called integrable on [a, b]).
Notation: In ∫ab f(x) dx, the number a is the lower limit, b is the upper limit, f(x) is the integrand, and dx indicates the variable of integration.
Integrability Theorem
If f is continuous on [a, b], or if f is bounded and has only finitely many discontinuities on [a, b], then f is integrable on [a, b].
Signed Area Interpretation
The definite integral computes signed area:
- Where f(x) ≥ 0, the integral counts area as positive.
- Where f(x) < 0, the integral counts area as negative.
Therefore ∫ab f(x) dx = (area above x-axis) − (area below x-axis).
Example 1: Signed Area Using Geometry
Evaluate ∫−22 x dx using geometry.
Solution: The graph of y = x is a line through the origin. On [−2, 0] it forms a triangle below the x-axis with area (1/2)(2)(2) = 2. On [0, 2] it forms a triangle above with area 2.
Signed area: 2 − 2 = 0.
This makes sense: by symmetry, the positive and negative areas cancel.
Example 2: Geometric Evaluation
Evaluate ∫03 (2x + 1) dx using geometry.
Solution: The graph is a line from (0, 1) to (3, 7). The region is a trapezoid with parallel sides 1 and 7, height 3.
Area = (1/2)(1 + 7)(3) = 12.
Properties of Definite Integrals
Key Properties
- ∫aa f(x) dx = 0
- ∫ba f(x) dx = −∫ab f(x) dx (reversing limits changes the sign)
- ∫ab c · f(x) dx = c · ∫ab f(x) dx (constant multiple)
- ∫ab [f(x) ± g(x)] dx = ∫ab f(x) dx ± ∫ab g(x) dx (sum/difference)
- ∫ab f(x) dx + ∫bc f(x) dx = ∫ac f(x) dx (additivity over intervals)
Example 3: Using Properties
Given ∫14 f(x) dx = 6 and ∫14 g(x) dx = −2, find ∫14 [3f(x) − 2g(x)] dx.
Solution: = 3 ∫14 f(x) dx − 2 ∫14 g(x) dx = 3(6) − 2(−2) = 18 + 4 = 22.
Example 4: Additivity
Given ∫05 f(x) dx = 10 and ∫03 f(x) dx = 7, find ∫35 f(x) dx.
Solution: By the additivity property: ∫03 f(x) dx + ∫35 f(x) dx = ∫05 f(x) dx.
So ∫35 f(x) dx = 10 − 7 = 3.
The Fundamental Theorem of Calculus, Part 1
This is one of the most important results in all of mathematics. It provides a practical way to evaluate definite integrals without computing limits of Riemann sums.
FTC Part 1 (Evaluation Theorem)
If f is continuous on [a, b] and F is any antiderivative of f (that is, F' = f), then:
∫ab f(x) dx = F(b) − F(a)
We write F(x) |ab = F(b) − F(a).
This is remarkable: instead of computing tedious limits of sums, we just need to find an antiderivative and plug in the endpoints.
Example 5: Applying FTC Part 1
Evaluate ∫13 x² dx.
Step 1: An antiderivative of x² is F(x) = x³/3.
Step 2: F(3) − F(1) = 27/3 − 1/3 = 9 − 1/3 = 26/3.
Example 6: FTC with Trig
Evaluate ∫0π/2 cos x dx.
Step 1: Antiderivative: F(x) = sin x.
Step 2: sin(π/2) − sin(0) = 1 − 0 = 1.
More Evaluation Theorem Examples
Example 7: Exponential Function
Evaluate ∫01 ex dx.
Solution: F(x) = ex. So ∫01 ex dx = e1 − e0 = e − 1 ≈ 1.718.
Example 8: Polynomial
Evaluate ∫−12 (3x² − 4x + 1) dx.
Step 1: F(x) = x³ − 2x² + x.
Step 2: F(2) − F(−1) = (8 − 8 + 2) − (−1 − 2 − 1) = 2 − (−4) = 6.
Comparison Properties
These inequalities are useful for estimating integrals:
- If f(x) ≥ 0 on [a, b], then ∫ab f(x) dx ≥ 0.
- If f(x) ≥ g(x) on [a, b], then ∫ab f(x) dx ≥ ∫ab g(x) dx.
- If m ≤ f(x) ≤ M on [a, b], then m(b − a) ≤ ∫ab f(x) dx ≤ M(b − a).
Key Takeaways
- The definite integral ∫ab f(x) dx is defined as the limit of Riemann sums.
- It computes signed area: positive above the x-axis, negative below.
- Properties like linearity and additivity allow you to break integrals apart and recombine them.
- FTC Part 1: ∫ab f(x) dx = F(b) − F(a) where F' = f. This is the evaluation shortcut.
Check Your Understanding
1. Evaluate ∫04 (x + 2) dx using the FTC.
2. Evaluate ∫0π sin x dx.
3. If ∫06 f(x) dx = 10 and ∫46 f(x) dx = 3, find ∫04 f(x) dx.
4. Evaluate ∫1e (1/x) dx.
Ready for More?
Next Lesson
In Lesson 4, you will learn FTC Part 2 (differentiating an integral) and the Net Change Theorem.
Start Lesson 4