Numerical Integration - Examples & Applications

This section explores methods for numerically estimating the definite integral of a function, including the Trapezoidal Rule, Simpson's Rules (1/3 and 3/8), integration of unequally spaced data, Romberg Integration, Gauss Quadrature, and Multiple Integrals.

Trapezoidal Rule

The trapezoidal rule approximates the area under a curve by connecting the endpoints of the interval with a straight line.

Basic: Single-Application Trapezoidal Rule

Estimate the integral of f(x)=0.2+25x−200x2+675x3−900x4+400x5f(x) = 0.2 + 25x - 200x^2 + 675x^3 - 900x^4 + 400x^5 from a=0a = 0 to b=0.8b = 0.8. The exact value of the integral is 1.6405331.640533. Calculate the true error.

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Intermediate: Multiple-Application Trapezoidal Rule

Estimate the same integral from a=0a=0 to b=0.8b=0.8 using a 2-segment multiple-application trapezoidal rule.

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Simpson's 1/3 and 3/8 Rules

Simpson's 1/3 rule fits a parabola through three points (n=2n=2), and Simpson's 3/8 rule fits a cubic equation through four points (n=3n=3).

Intermediate: Simpson's 1/3 Rule

Estimate the same integral of f(x)=0.2+25x−200x2+675x3−900x4+400x5f(x) = 0.2 + 25x - 200x^2 + 675x^3 - 900x^4 + 400x^5 from a=0a=0 to b=0.8b=0.8 using Simpson's 1/3 rule. The true value is 1.6405331.640533.

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Advanced: Simpson's 3/8 Rule

Estimate the integral of the same function from a=0a=0 to b=0.8b=0.8 using Simpson's 3/8 rule (n=3n=3 segments).

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Integration with Unequal Segments

When data points are unequally spaced, the trapezoidal rule is applied individually to each segment and the results are summed.

Intermediate: Unequal Segments Trapezoidal Rule

Calculate the integral of the data: (x,y)(x, y): (0,0.2),(0.1,1.3),(0.4,2.5),(0.8,0.2)(0, 0.2), (0.1, 1.3), (0.4, 2.5), (0.8, 0.2)

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Romberg Integration

Romberg integration combines two trapezoidal rule estimates (often derived using hh and h/2h/2) using Richardson extrapolation to obtain a more accurate estimate.

Advanced: Romberg Integration

Given the trapezoidal rule estimates for h1=0.4h_1 = 0.4 (I(h1)=1.0688I(h_1) = 1.0688) and h2=0.2h_2 = 0.2 (I(h2)=1.4848I(h_2) = 1.4848), use Romberg integration to calculate an O(h4)O(h^4) estimate.

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Gauss Quadrature

Gauss quadrature optimizes both the weights and the function evaluation points to achieve the highest possible accuracy for a given number of points. It evaluates integrals on the standard domain [−1,1][-1, 1].

Advanced: Two-Point Gauss-Legendre Quadrature

Evaluate the integral of f(x)=0.2+25x−200x2+675x3−900x4+400x5f(x) = 0.2 + 25x - 200x^2 + 675x^3 - 900x^4 + 400x^5 from a=0a = 0 to b=0.8b = 0.8 using two-point Gauss-Legendre quadrature. The true value is 1.6405331.640533.

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Multiple Integrals

Multiple integrals are solved by applying one-dimensional integration techniques sequentially.

Advanced: Double Integral using Trapezoidal Rule

Evaluate the double integral of f(x,y)=x2yf(x,y) = x^2y over the rectangular region x∈[0,2],y∈[1,3]x \in [0, 2], y \in [1, 3] using the single-application trapezoidal rule in both directions.

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