VG3
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Quiz
What is integral calculus?
Integral calculus is the second major branch of calculus — the reverse of differentiation. It is about areas, volumes and total change.
🎮 Try it yourself — interactive!
Choose a function and drag the limits. Watch the area under the curve fill up live.
→ Open interactive integral
Choose a function and drag the limits. Watch the area under the curve fill up live.
→ Open interactive integral
The idea
While differentiation gives us instantaneous change, integration sums up infinitely many infinitely small contributions — giving the area under a curve.
The indefinite integral
∫ f(x) dx = F(x) + C
F(x) is called the antiderivative, C is the constant of integration.
Integration rules
| f(x) | ∫ f(x) dx | Example |
|---|---|---|
| xⁿ (n ≠ -1) | xⁿ⁺¹/(n+1) + C | ∫x² dx = x³/3 + C |
| k (constant) | kx + C | ∫5 dx = 5x + C |
| eˣ | eˣ + C | ∫eˣ dx = eˣ + C |
| 1/x | ln|x| + C | ∫(1/x) dx = ln|x| + C |
The definite integral
∫ₐᵇ f(x) dx = F(b) - F(a)
Example
Calculate ∫₀² x² dx
Antiderivative: F(x) = x³/3
F(2) - F(0) = 8/3 - 0 = 8/3 ≈ 2.67
F(2) - F(0) = 8/3 - 0 = 8/3 ≈ 2.67
Fundamental theorem of calculus:
Differentiation and integration are inverse operations — like multiplication and division.
Differentiation and integration are inverse operations — like multiplication and division.
Applications
- Calculating areas and volumes
- Finding distance from a velocity function
- Work in physics
- Probability distributions
Give me a place to stand, and I shall move the earth.
— Archimedes (287–212 BC)
🧠 Test yourself
Question 1 of 3
What is the integral of x2?
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