3.6 Numerical integration  (Page 2/11)

The trapezoidal rule

We can also approximate the value of a definite integral by using trapezoids rather than rectangles. In [link] , the area beneath the curve is approximated by trapezoids rather than by rectangles.

The trapezoidal rule for estimating definite integrals uses trapezoids rather than rectangles to approximate the area under a curve. To gain insight into the final form of the rule, consider the trapezoids shown in [link] . We assume that the length of each subinterval is given by $\text{Δ}x.$ First, recall that the area of a trapezoid with a height of h and bases of length $b_1$ and $b_2$ is given by $\text{Area}=\frac{1}{2}h(b_1+b_2).$ We see that the first trapezoid has a height $\text{Δ}x$ and parallel bases of length $f(x_0)$ and $f(x_1).$ Thus, the area of the first trapezoid in [link] is

The areas of the remaining three trapezoids are

Consequently,

After taking out a common factor of $\frac{1}{2}\text{Δ}x$ and combining like terms, we have

Generalizing, we formally state the following rule.

Before continuing, let’s make a few observations about the trapezoidal rule. First of all, it is useful to note that

That is, $L_n$ and $R_n$ approximate the integral using the left-hand and right-hand endpoints of each subinterval, respectively. In addition, a careful examination of [link] leads us to make the following observations about using the trapezoidal rules and midpoint rules to estimate the definite integral of a nonnegative function. The trapezoidal rule tends to overestimate the value of a definite integral systematically over intervals where the function is concave up and to underestimate the value of a definite integral systematically over intervals where the function is concave down. On the other hand, the midpoint rule tends to average out these errors somewhat by partially overestimating and partially underestimating the value of the definite integral over these same types of intervals. This leads us to hypothesize that, in general, the midpoint rule tends to be more accurate than the trapezoidal rule.

Absolute and relative error

An important aspect of using these numerical approximation rules consists of calculating the error in using them for estimating the value of a definite integral. We first need to define absolute error    and relative error    .