Calculus › Section I · Integration techniques and improper integrals › Topic I.6
Numerical integration: trapezoid and Simpson's rules
Most definite integrals have no antiderivative you can write down — e^(x²), sin(x²), eˣ/x all resist every technique in this chapter. But a definite integral is just a number, and numbers can be approximated: slice the region into simple shapes and add their areas. The trapezoid rule uses straight tops; Simpson's rule fits parabolas, and the accuracy jumps dramatically.
In this topic
- The trapezoid rule
- Worked example: trapezoid on a known area
- Simpson's rule: parabolas instead
- Worked example: Simpson on the same integral
- How the errors behave
- Worked example: an impossible antiderivative
- When numerical rules are the only option
- Common mistakes to avoid
- Key takeaways
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Standards: 6