Calculus & Analysis Difficulty: Introductory

Integration

Integration generalizes area under a curve and is the inverse operation of differentiation (Fundamental Theorem of Calculus). It is essential for probability, physics, and signal processing.

Key Points

  • The Riemann integral approximates area via finite sums.
  • The Lebesgue integral generalizes integration to a broader class of functions.
  • The Fundamental Theorem connects differentiation and integration.

Formulas

Definite integral
$$\int_a^b f(x) \, dx = \lim_{\|P\| \to 0} \sum_{i=1}^n f(x_i^*) \Delta x_i$$
Fundamental Theorem of Calculus
$$\frac{d}{dx} \int_a^x f(t) \, dt = f(x)$$

Code Example

from scipy import integrate

def f(x):
    return x ** 2

result, error = integrate.quad(f, 0, 1)
print(result)  # 0.333...

Tags

  • foundations
  • area
  • measure

References

  • Calculus
    Michael Spivak · Publish or Perish · source
  • Real Analysis
    H.L. Royden and P.M. Fitzpatrick · Pearson · source

Knowledge Graph