Simpson’s Rule
Approximate area with parabolas.
Key idea
Approximate area with parabolas.
In depth
Simpson fits parabolas through three points at a time: S = (Δx/3)(f₀ + 4f₁ + 2f₂ + 4f₃ + ... + fₙ), n even. Error shrinks like 1/n⁴, and it is exact for cubics.
Worked example
∫₀² x² dx with n = 2: (1/3)(0 + 4 + 4) = 8/3, the exact value.
Formula
Real-world uses
- Estimating integrals in engineering software and calculators with high accuracy
Common mistake
Using an odd number of intervals.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Simpson’s Rule?
Approximate area with parabolas. It is a topic at the expert level in Calculus.
What is the formula for Simpson’s Rule?
The key formula is (h/3)[f₀+4f₁+2f₂+…+fₙ].
Where is Simpson’s Rule used in real life?
Real-world examples: Estimating integrals in engineering software and calculators with high accuracy.
What should I learn before Simpson’s Rule?
Make sure you are comfortable with Trapezoid Rule for Integrals, the previous topic in Calculus.
What is a common mistake with Simpson’s Rule?
Using an odd number of intervals.
How can I practise Simpson’s Rule?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Simpson’s Rule?
Move on to Sequences and Limits next.
Related expert topics in Calculus
- Derivatives of Inverse Trig – Derivatives of arcsin, arccos, arctan.
- Implicit Differentiation – Differentiate both sides of F(x,y)=0 treating y as a function of x.
- Logarithmic Differentiation – Take logs before differentiating products and powers.
- Related Rates – Link rates of change of connected quantities.
- Linear Approximation – Approximate f near a using its tangent line.
- Differentials – Use dy = f′(x)dx for small changes.