Two kinds of integrals
| Indefinite integral | Definite integral |
|---|---|
| ∫ f(x) dx | ∫ₐᵇ f(x) dx |
| a family of functions + C | a single number |
| reverses differentiation | measures accumulated change or net area |
Evaluating a definite integral
The Fundamental Theorem of Calculus links the two:
∫ₐᵇ f(x) dx = F(b) − F(a), where F is any antiderivative of f.
The numbers a and b are the limits of integration.
Example: ∫₁³ 2x dx
- F(x) = x²
- F(3) − F(1) = 9 − 1 = 8
(The + C cancels, so it’s left out.)
More examples
- ∫₀² (3x² + 1) dx = [x³ + x]₀² = (8 + 2) − 0 = 10
- ∫₁⁴ √x dx = [(2/3)x^(3/2)]₁⁴ = (2/3)(8) − (2/3)(1) = 14/3
- ∫₀^π sin x dx = [−cos x]₀^π = (1) − (−1) = 2
Where the idea comes from: Riemann sums
To find the area under a curve, slice it into thin rectangles and add their areas. This is a Riemann sum.
Example: Area under y = x² from 0 to 2 with 4 right-endpoint rectangles (width 0.5):
0.5(0.25 + 1 + 2.25 + 4) = 3.75
As the rectangles get thinner, the sum approaches the exact value: ∫₀² x² dx = 8/3 ≈ 2.67. (Right endpoints overestimate for an increasing function.)
Properties
| Property | Rule |
|---|---|
| reversing limits | ∫ₐᵇ f = −∫ᵇₐ f |
| same limits | ∫ₐᵃ f = 0 |
| splitting | ∫ₐᵇ f + ∫ᵇᶜ f = ∫ₐᶜ f |
| constants | ∫ k f = k ∫ f |
Net change
Integrating a rate gives the net change:
∫ₐᵇ (rate) dt = total change from a to b
Example: Water flows into a tank at r(t) = 2t + 3 L/min. How much enters in the first 5 minutes?
∫₀⁵ (2t + 3) dt = [t² + 3t]₀⁵ = 25 + 15 = 40 L
Example: Velocity v(t) = 6 − 2t. Displacement from t = 0 to 5: [6t − t²]₀⁵ = 30 − 25 = 5 m.
Signed area
Area below the x-axis counts as negative in a definite integral. To find total area, split the integral where the graph crosses the axis and add absolute values.
Worked examples
Example 1: ∫₂⁵ 4 dx = 4(5 − 2) = 12.
Example 2: ∫₀¹ x³ dx = 1/4.
Example 3: ∫₀² (x − 1) dx = [x²/2 − x]₀² = 0 → net area 0 (equal areas above and below).
Common mistakes
- Subtracting in the wrong order (F(a) − F(b)).
- Adding + C to a definite integral answer.
- Forgetting negative area below the axis.
Try it yourself
- ∫₁² 3x² dx
- ∫₀⁴ (x + 2) dx
- ∫₀^(π/2) cos x dx
- A car’s velocity is v(t) = 3t² m/s. Distance in the first 2 s?
Answers: 1) 7 2) 16 3) 1 4) 8 m
Quick recap
- A definite integral gives a number; an indefinite integral gives a function + C.
- Fundamental Theorem of Calculus: ∫ₐᵇ f = F(b) − F(a).
- Integrating a rate gives the net change; Riemann sums explain why.