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What is Integration?

Integration is the process of finding the total accumulation of something — the area under a curve, the total distance traveled, the total work done. It is the reverse operation of differentiation and the second major pillar of calculus.

In this lesson

  1. The intuition: adding up infinitely many pieces
  2. Definite integrals and area
  3. Indefinite integrals and antiderivatives
  4. Basic integration rules
  5. Real-world applications

1 The Intuition: Adding Up Infinitely Many Pieces

Suppose you want to find the area under a curve y = f(x) between x=a and x=b. You could approximate it with rectangles: divide the interval into n thin slices, build a rectangle on each, and sum the areas. The more rectangles you use, the better the approximation.

Integration takes this to its limit: use infinitely many infinitely thin rectangles. Each has width dx (an infinitesimal) and height f(x), contributing f(x)dx to the sum. The integral symbol ∫ is an elongated S standing for "sum." The definite integral ∫ₐᵇ f(x)dx is the limit of this sum as the number of rectangles approaches infinity.

The Riemann Sum

The formal definition: ∫ₐᵇ f(x)dx = lim(n→∞) Σᵢ f(xᵢ*)Δx. This is a sum of n rectangle areas, taken to its limit as n→∞ and Δx→0. This limit exists (and equals the integral) for any continuous function.

2 Definite Integrals and Area

∫ₐᵇ f(x)dx gives the signed area between f(x) and the x-axis from x=a to x=b. "Signed" means regions below the x-axis contribute negative area.

Interpreting a Definite Integral
∫₀³ x dx — area under y=x from 0 to 3
1The region is a right triangle with base 3 and height 3
2Area = ½ × base × height = ½ × 3 × 3 = 4.5
3The integral equals 4.5 — confirmed by calculation
Answer: ∫₀³ x dx = 4.5

3 Indefinite Integrals and Antiderivatives

An antiderivative of f(x) is a function F(x) such that F'(x) = f(x). The indefinite integral ∫f(x)dx = F(x) + C, where C is the constant of integration (any constant has zero derivative, so there are infinitely many antiderivatives).

Example: ∫2x dx = x² + C, because d/dx(x²+C) = 2x. The +C is essential — without it, you're missing infinitely many valid answers.

4 Basic Integration Rules

Core integration rules
∫∫xⁿ dx = xⁿ⁺¹/(n+1) + C (Power rule — reverse of differentiation power rule, n ≠ −1)
∫∫k dx = kx + C (constant integrates to kx)
∫∫[f(x) + g(x)] dx = ∫f(x)dx + ∫g(x)dx (sum rule)
∫∫cf(x) dx = c∫f(x)dx (constant multiple rule)
Applying Integration Rules
Find ∫(3x² + 4x − 1) dx
1Apply power rule term by term:
2∫3x² dx = 3 · x³/3 = x³
3∫4x dx = 4 · x²/2 = 2x²
4∫(−1) dx = −x
5Result: x³ + 2x² − x + C
Answer: ∫(3x² + 4x − 1) dx = x³ + 2x² − x + C

5 Real-World Applications

Physics: if velocity is v(t), then total displacement = ∫v(t)dt. If acceleration is a(t), then velocity = ∫a(t)dt. Integration converts rates of change back into accumulated quantities.

Engineering: structural loads, electrical charge (integral of current), fluid flow, and heat transfer all involve integration. The center of mass of an irregular object is found by integration.

Probability: for continuous random variables, the probability of a range of outcomes is the integral of the probability density function. The normal distribution's probabilities come from integrating the bell curve formula.

Practice Problems

Find ∫(6x² − 4x + 2) dx
Power rule: 6x³/3 − 4x²/2 + 2x + C = 2x³ − 2x² + 2x + C
Find ∫x⁴ dx
Power rule: x⁵/5 + C = x⁵/5 + C
Interpret: ∫₀² 3 dx geometrically
This is the area of a rectangle with height 3 and width 2 = 6. The integral of a constant k from a to b equals k(b−a).