Numerical Integration: Trapezoidal & Simpson's Rules

≈ 25 min

When an integral $\int_a^b f(x)\, dx$ has no elementary antiderivative, we approximate it.

Trapezoidal rule (single application): replace the curve by a straight line across $[a, b]$ :

\int_a^b f\, dx \approx \frac{b - a}{2}\big(f(a) + f(b)\big).

With $n$ equal strips of width $h = \tfrac{b-a}{n}$ : $\;\dfrac{h}{2}\big(f_0 + 2f_1 + \cdots + 2f_{n-1} + f_n\big).$

Simpson's rule (needs an even number of strips) fits parabolas instead of straight lines:

\frac{h}{3}\big(f_0 + 4f_1 + 2f_2 + 4f_3 + \cdots + f_n\big).

Worked example. For $\int_0^2 x^2\, dx$ (exact value $\tfrac{8}{3} \approx 2.667$ ): Simpson with $n = 2$ gives $\tfrac{1}{3}(0 + 4(1) + 4) = \tfrac{8}{3}$ — exact, because Simpson is perfect for quadratics.

Simpson's rule requires the number of intervals (strips) to be…

even

odd

prime

a multiple of 3

Estimate $\int_0^2 x^2 \, dx$ using the trapezoidal rule with a single application (one strip).

Estimate $\int_0^2 x^2 \, dx$ using the composite trapezoidal rule with $n = 2$ strips ( $h = 1$ ).

Estimate $\int_0^2 x^2 \, dx$ using Simpson's rule with $n = 2$ ( $h = 1$ ). Give your answer to 3 decimal places.