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Integral

Formula

\[ \int_a^b f(x)\,dx \]
\[ \frac{d}{dx}\int^x f(t)\,dt = f(x) \]

Parameters

  • \(f(x)\): integrand
  • \([a,b]\): interval of integration

What it means

An integral accumulates quantities (often area under a curve) over an interval.

What it's used for

  • Areas, accumulated change, and continuous probability.
  • Solving differential and physical models.

Key properties

  • Linear operator.
  • Linked to derivatives by the Fundamental Theorem of Calculus.

Common gotchas

  • Indefinite and definite integrals are different objects.
  • Bounds and variable of integration are easy to mishandle.

Example

\(\int_0^1 x\,dx = 1/2\).

How to Compute (Pseudocode)

Input: function f, interval [a,b], numerical integration method, subdivisions n
Output: approximate integral value

# Example: composite trapezoidal rule
h <- (b - a) / n
total <- 0.5 * f(a) + 0.5 * f(b)
for i from 1 to n-1:
  total <- total + f(a + i*h)
return h * total

Complexity

  • Time: \(O(n)\) function evaluations for common fixed-grid numerical rules (for example, trapezoidal or midpoint)
  • Space: \(O(1)\) extra space
  • Assumptions: Numerical approximation of a definite integral is shown; symbolic integration uses different algorithms/workflows

See also