Integrals and Accumulation

1. Area and Accumulation

Definite integral accumulates infinitesimal contributions:

integral_a^b f(x) dx.

Geometrically: signed area under curve.

2. Riemann Sum View

Partition interval into small pieces:

sum f(x_i^*) Delta x.

Integral is limit as max interval width goes to zero.

3. Fundamental Theorem of Calculus

If F'(x)=f(x), then:

integral_a^b f(x) dx = F(b)-F(a).

This links differentiation and integration.

4. Improper Integrals

For infinite intervals or singularities, define integral by limits. Convergence must be checked explicitly.

5. Numerical Integration

When antiderivative unavailable: - Trapezoidal rule: O(h^2) global error - Simpson’s rule: O(h^4) global error - Monte Carlo integration: dimension-friendly stochastic approach

6. Worked Example

Compute integral_0^1 x^2 dx = 1/3.

Trapezoid with two intervals gives approximation 0.375, showing discretization error.

7. CS Applications

• probability from densities: P(a<=X<=b)=integral_a^b f_X(x)dx
• total energy or mass in simulation
• accumulated loss/utility over time

Exercises

1. Derive trapezoid rule from linear interpolation.
2. Compare trapezoid vs Simpson on sin(x) over [0,pi].
3. Use Monte Carlo to estimate area of unit circle.
4. Explain when Monte Carlo can beat grid methods.