Which functions does Simpson's rule integrate exactly according to this lecture?

Conditions

• The method fits three points to a parabola and integrates that parabola.
• Applies to constant, linear, and quadratic functions as stated in the lecture.

Reasoning, step by step

1. Identify the approximating curve as a parabola.
2. Recognize that a parabola is a quadratic polynomial.
3. Conclude that the method is exact for polynomials of degree up to two (constants, linear, and parabolas).

Example

The speaker states an important exactness property: because the approximating curve is quadratic, Simpson’s rule integrates exactly any function that is constant, linear, or a parabola.

Common misconceptions

• Assuming Simpson's rule is exact for cubic or higher-degree polynomials.
• Thinking that Simpson's rule is only an approximation and never exact.

Watch the explanation

Explore next

• What is Romburg integration in relation to Simpson's rule?
• How is Gaussian integration different from the trapezoidal and Simpson methods mentioned here?