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MathidiaMathematics Academy
Numerical methods
Lesson 59 of 75
Challenge

Numerical integration

Objective. Apply trapezium and Simpson rules with error awareness.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

Numerical integration approximates ∫ₐᵇ f(x)dx when an antiderivative is hard to find, using rules like the trapezium rule, which approximates the area under each subinterval by a trapezium: T = (h/2)(f(x₀) + 2f(x₁) + ... + f(xₙ)), with h=(b−a)/n. Simpson's rule fits parabolas instead, giving S = (h/3)(f(x₀) + 4f(x₁) + 2f(x₂) + ... + f(xₙ)) for even n, which is more accurate for smooth functions. For ∫₀² x² dx with n=2 (h=1), the trapezium rule gives (1/2)(0 + 2(1) + 4) = 3, close to the exact value 8/3.

Worked example

Use the trapezium rule with n=2 strips to approximate ∫₀² x dx as an example of numerical integration.

2
Illustrated card showing 2
  1. 1Step size: h = (2−0)/2 = 1
  2. 2Evaluate f(0)=0, f(1)=1, f(2)=2
  3. 3Trapezium rule: T = (h/2)(f(0)+2f(1)+f(2))
  4. 4Compute: T = (1/2)(0+2+2) = 2

Answer: 2

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Common mistake. Applying Simpson's rule with an odd number of subintervals; Simpson's rule requires an even number of subintervals (odd number of points) to pair them into parabolic panels correctly.

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