AdvancedTopic 94 of 104Engineering Mathematics
Numerical integration
Trapezoidal and Simpson rules approximate integrals from samples.
In depth
The trapezoidal rule approximates area with trapezoids (error ∝ h²). Simpson's rule fits parabolas through three points (error ∝ h⁴). Gaussian quadrature uses optimal nodes. Adaptive methods refine where the function changes fast.
Key points
- Composite rules sum subintervals
- Richardson extrapolation improves accuracy
- Monte Carlo for high dimensions
ExampleTotal energy from sampled power readings is found by trapezoidal integration.
What is Numerical integration?
Trapezoidal and Simpson rules approximate integrals from samples.
Why is it important to understand numerical integration?
Numerical integration lets you compute areas and totals from measured data, which is common in test and energy analysis.
When is Simpson's rule better than the trapezoidal rule?
Simpson's rule is more accurate for smooth functions because it fits parabolas. It needs an even number of intervals.
What causes errors in numerical integration?
Coarse steps and rapidly changing functions. Refining the step size and checking convergence reveal whether the result is trustworthy.
How is Numerical integration used in practice?
Typical applications include finding the energy used from a logged power profile, computing the area of an irregular cross-section from survey points and estimating the total flow from a sampled flow-rate curve.
What should I learn before Numerical integration?
Study "Numerical differentiation" first (topic 93 of 104 in Engineering Mathematics). It covers the groundwork this topic builds on.
What are the key points of Numerical integration?
Composite rules sum subintervals. Richardson extrapolation improves accuracy. Monte Carlo for high dimensions.
Can you give an example of Numerical integration?
Total energy from sampled power readings is found by trapezoidal integration.
What comes after Numerical integration?
The next topic is "Solving ODEs numerically", which builds directly on this one.
How difficult is Numerical integration?
It is classed as Advanced in Engineering Mathematics (topic 94 of 104). It is an advanced topic. It assumes you are comfortable with the intermediate topics in Engineering Mathematics.