The trapezoidal rule

An integral whose antiderivative cannot be found is approximated numerically. Laying out trapezoids instead of rectangles is the trapezoidal rule¹. We look at its accuracy for $f(x) = x^2$ from $0$ to $2$ .

Trying two pieces

The area of a trapezoid is the average of the two parallel sides times the width.

\frac{f(0) + f(1)}{2} \times 1 + \frac{f(1) + f(2)}{2} \times 1 = \frac{0 + 1}{2} + \frac{1 + 4}{2} = 3

The true value is $\dfrac{8}{3} \approx 2.667$ , so the error is $\dfrac{1}{3}$ .

The general form

It is a sum with half weight on the two ends.

\int_a^b f(x)\,dx \approx h \left( \frac{f_0}{2} + f_1 + \cdots + f_{n-1} + \frac{f_n}{2} \right)

It is also the average of the left-endpoint sum and the right-endpoint sum.

Construction	Value
Left-endpoint sum	$1$
Right-endpoint sum	$5$
Their average, the trapezoidal rule	$3$

An underestimate and an overestimate, added and halved.

Refining the partition

Partition	$h$	Approximation	Error
Two pieces	$1$	$3$	$\dfrac{1}{3}$
Four pieces	$0.5$	$2.75$	$\dfrac{1}{12}$

Halving $h$ quarters the error, which shows that the error is proportional to $h^2$ . The error of a plain rectangle sum is proportional to $h$ , so the trapezoidal rule shrinks faster.

Which way the error goes

Concavity of the function	What the trapezoidal rule gives
Concave up	an overestimate
Concave down	an underestimate

It is decided by whether the top of the trapezoid lies above or below the curve. Knowing the concavity tells one the direction of the error.

Improving the accuracy further

Approximate with parabolas rather than lines. Joining three points at a time with a parabola gives Simpson's rule, whose error is proportional to $h^4$ , and with the same four pieces it reproduces $\dfrac{8}{3}$ exactly². A quadratic is represented perfectly by a parabola.

The smooth curve on the graph is $y = x^2$ , the polyline is the approximation joining the tops of the trapezoids, and the large dots are the division points.

Trapezoidal rule, Wikipedia
Simpson's rule, Wikipedia