Numerical Integration (Trapezoidal And Simpsons Rules)

1. Core Concept Overview & Intuition

Objectives of the Lesson

By the end of this lesson you should be able to:

• explain why numerical integration is needed when an antiderivative is unavailable or when the integrand is known only through numerical values;
• interpret the definite integral as a signed area and as a limit of sums;
• derive and apply the area formula for a trapezium;
• state and apply the Trapezoidal Rule to approximate a definite integral;
• explain the parabolic-fitting principle behind Simpson’s Rule;
• state and apply Simpson’s Rule when the number of subintervals is even;
• compare the two rules and describe how accuracy is affected by refining the partition.

Why Numerical Integration Is Needed

The definite integral

$$
\int_a^b f(x)\,dx
$$

represents the signed area between the curve $y=f(x)$ and the $x$-axis from $x=a$ to $x=b$. In earlier work, such integrals were evaluated exactly by finding an antiderivative and applying the Fundamental Theorem of the Integral Calculus. This exact approach is not always available. The antiderivative of the integrand may have no elementary expression, the integrand may be known only at a finite set of measured or computed points, or evaluating the antiderivative may be impractical. Numerical integration provides systematic approximations based on values of $f$ at selected points.

The central idea is to divide the interval $[a,b]$ into smaller subintervals, approximate the contribution from each subinterval by a simple geometric shape, and add the contributions. The Trapezoidal Rule uses straight-line segments and trapezia. Simpson’s Rule uses parabolic arcs and gives greater accuracy for smooth functions.

Equal Subintervals and Ordinates

To apply either rule, divide $[a,b]$ into $n$ equal subintervals. The common width is

$$
h=\frac{b-a}{n}.
$$

The division points are

$$
\begin{aligned}
x_0 &= a,\
x_1 &= a+h,\
x_2 &= a+2h,\
&\vdots\
x_n &= a+nh=b.
\end{aligned}
$$

The values

$$
y_i=f(x_i),\qquad i=0,1,\ldots,n,
$$

are called the ordinates. The first and last ordinates are $y_0=f(a)$ and $y_n=f(b)$. The interior ordinates are $y_1,y_2,\ldots,y_{n-1}$. A smaller value of $h$ means more subintervals and generally a better approximation, at the cost of more function evaluations.

Area of a Trapezium

A trapezium is a plane figure with one pair of parallel sides. If the parallel sides have lengths $p$ and $q$, and the perpendicular distance between them is $h$, then the area is

$$
A=\frac{1}{2}(p+q)h.
$$

Equivalently, the area is the height multiplied by the arithmetic mean of the parallel sides. In a single strip under the curve $y=f(x)$, the two parallel sides are the consecutive ordinates $y_{i-1}$ and $y_i$, and the height is the subinterval width $h$. Thus the strip area is approximated by

$$
A_i=\frac{1}{2}(y_{i-1}+y_i)h.
$$

Trapezoidal Rule Intuition

The Trapezoidal Rule replaces the curve on each subinterval $[x_{i-1},x_i]$ by the straight chord joining the points $(x_{i-1},y_{i-1})$ and $(x_i,y_i)$. Each strip is therefore treated as a trapezium. Adding the areas of all strips gives

$$
T_n=\frac{h}{2}\left[y_0+2(y_1+y_2+\cdots+y_{n-1})+y_n\right].
$$

The first and last ordinates appear once, while every interior ordinate appears twice because each interior ordinate is shared by two adjacent trapezia. This is the composite Trapezoidal Rule. It is simple, robust, and applicable for any positive integer $n$.

Simpson’s Rule Intuition

Simpson’s Rule improves on the straight-line approximation by fitting a parabola through each group of three consecutive points. Since three points determine a parabola, the method works two adjacent subintervals at a time. Consequently, the number of subintervals must be even. The parabola captures the curvature of $f$ over each pair of strips, so Simpson’s Rule is usually more accurate than the Trapezoidal Rule for smooth functions.

The weighting pattern produced by the parabolic fitting is distinctive. The end ordinates receive weight $1$, the odd-numbered interior ordinates receive weight $4$, and the even-numbered interior ordinates receive weight $2$. The multiplier outside the bracket is $\frac{h}{3}$ rather than $\frac{h}{2}$.

Practical Interpretation

Both rules return approximations, not exact values in general. Refining the partition by increasing $n$ usually improves the approximation. The Trapezoidal Rule is second-order accurate, while Simpson’s Rule is fourth-order accurate for sufficiently smooth functions. Simpson’s Rule is often preferred when $f$ is smooth and an even number of subintervals can be used; the Trapezoidal Rule remains useful when the data are limited or when a simple, stable method is required.

2. Mathematical Formalization & Core Definitions

Uniform Partition and Notation

> Definition (Uniform Partition). Let $f$ be defined on $[a,b]$, and let $n$ be a positive integer. Put $h=(b-a)/n$ and $x_i=a+ih$ for $i=0,1,\ldots,n$. The points $x_0,x_1,\ldots,x_n$ form a uniform partition of $[a,b]$ with step size $h$, and the values $y_i=f(x_i)$ are called the ordinates.

The Definite Integral as a Limit of Sums

For a continuous function $f$ on $[a,b]$, the definite integral is the limit of Riemann sums:

$$
\int_a^b f(x)\,dx=\lim_{n\to\infty}\sum_{i=1}^{n}f(\xi_i)(x_i-x_{i-1}),
$$

where $\xi_i\in[x_{i-1},x_i]$ for each $i$. For a uniform partition, this becomes

$$
\int_a^b f(x)\,dx=\lim_{n\to\infty}\sum_{i=1}^{n}f(\xi_i)h.
$$

The numerical rules replace the limit by a finite sum and replace $f(\xi_i)$ by weighted combinations of the ordinates.

Trapezoidal Rule Formula

> Definition (Composite Trapezoidal Rule). Let $f$ be continuous on $[a,b]$, let $n$ be a positive integer, put $h=(b-a)/n$, and let $y_i=f(x_i)$ for $i=0,1,\ldots,n$. The composite Trapezoidal Rule approximation is

$$
> T_n=\frac{h}{2}\left[y_0+2\sum_{i=1}^{n-1}y_i+y_n\right].
$$

The formula is obtained by summing the areas of the trapezia:

$$
T_n=\sum_{i=1}^{n}\frac{h}{2}(y_{i-1}+y_i).
$$

Expanding the sum gives

$$
T_n=\frac{h}{2}\left[y_0+2y_1+2y_2+\cdots+2y_{n-1}+y_n\right].
$$

Thus the end ordinates are counted once and the interior ordinates are counted twice.

Simpson’s Rule Formula

> Definition (Composite Simpson’s Rule). Let $f$ be continuous on $[a,b]$, let $n$ be an even positive integer, put $h=(b-a)/n$, and let $y_i=f(x_i)$ for $i=0,1,\ldots,n$. The composite Simpson’s Rule approximation is

$$
> S_n=\frac{h}{3}\left[y_0+4\sum_{i=1,\,i\ \text{odd}}^{n-1}y_i+2\sum_{i=2,\,i\ \text{even}}^{n-2}y_i+y_n\right].
$$

Equivalently, writing the ordinates explicitly gives

$$
S_n=\frac{h}{3}\left[y_0+4(y_1+y_3+\cdots+y_{n-1})+2(y_2+y_4+\cdots+y_{n-2})+y_n\right].
$$

The rule uses pairs of adjacent subintervals. On each pair, a parabola is fitted through the three ordinates at the endpoints and midpoint of the pair. The condition that $n$ be even ensures that the whole interval $[a,b]$ can be partitioned into such pairs.

Accuracy and Error Remarks

For sufficiently smooth $f$, the composite Trapezoidal Rule has an error term of the form

$$
E_T=-\frac{(b-a)h^2}{12}f''(\xi)
$$

for some $\xi\in[a,b]$. The composite Simpson’s Rule has an error term of the form

$$
E_S=-\frac{(b-a)h^4}{180}f^{(4)}(\eta)
$$

for some $\eta\in[a,b]$. These formulas indicate that the Trapezoidal Rule is second-order accurate, with error proportional to $h^2$, while Simpson’s Rule is fourth-order accurate, with error proportional to $h^4$, for functions with enough continuous derivatives.