THE DEFINITE INTEGRAL THE MIDPOINT RULE In the previous sections, we found that limits of sums could be computed using sample endpoints (which include right endpoints, left endpoints, or any number within the subinterval). In most cases, we choose to use right endpoints because it is considerably convenient for computing limits of sums. If, however, our goal is to find an estimate (i.e. an approximation) of a definite integral, it is better to choose the sample point to be the midpoint of the interval, which is denoted by xi. Note that ANY definite integral could be estimated using the Riemann sum, but if we decide to use midpoints, we follow this:
∫
b a
n
f(x) dx
∑
=
i=1
f(xi ) ∆x
=
∆x [f(x1) + f(x2) + ........ + f(xn)]
Where ∆x = (b – a)/n, and
xi
=
½ [xi-1
+
xi] is the midpoint of the subinterval [xi-1, xi]
So now, we’ll illustrate the use of the midpoint rule in the following examples. (In each example, we’ll use the given value of n to estimate the integral, which will then be rounded to four decimal places). Note that, when we use the midpoint rule, we are only making approximations. Therefore, we do not always know accurate our estimates are. We will, however, learn a method used for estimating the error involved while making an approximation using the midpoint rule. We will also learn a more efficient method for approximating definite integrals.
Example 1 ∫
10 0
(sin √x ) dx
n=5
SOLUTION Here, we have 5 subintervals, whose right end points, are 0,2,4,6,8 and 10. The figure below the graph of f(x) = sin √x plotted on the interval [0,10]: