T n = 1 2 ( L n + R n ) where L n = ∑ i = 1 n f ( x i − 1 ) Δ x and R n = ∑ i = 1 n f ( x i ) Δ x. Recall that a Riemann sum of a function f ( x ) f ( x ) over an interval is obtained by selecting a partition In general, any Riemann sum of a function f ( x ) f ( x ) over an interval may be viewed as an estimate of ∫ a b f ( x ) d x.
The Midpoint RuleĮarlier in this text we defined the definite integral of a function over an interval as the limit of Riemann sums. In addition, we examine the process of estimating the error in using these techniques. In this section we explore several of these techniques. Consequently, rather than evaluate definite integrals of these functions directly, we resort to various techniques of numerical integration to approximate their values. The antiderivatives of many functions either cannot be expressed or cannot be expressed easily in closed form (that is, in terms of known functions). 3.6.5 Use Simpson’s rule to approximate the value of a definite integral to a given accuracy.3.6.4 Recognize when the midpoint and trapezoidal rules over- or underestimate the true value of an integral.3.6.3 Estimate the absolute and relative error using an error-bound formula.3.6.2 Determine the absolute and relative error in using a numerical integration technique.3.6.1 Approximate the value of a definite integral by using the midpoint and trapezoidal rules.
