   Chapter 7.R, Problem 5CC

Chapter
Section
Textbook Problem

# State the rules for approximating the definite integral ∫ a b f ( x )   d x with the Midpoint Rule, the Trapezoidal Rule, and Simpson’s Rule. Which would you expect to give the best estimate? How do you approximate the error for each rule?

To determine

To Find: Rules for approximating definite integrals abf(x)dx with Midpoint Rule, Trapezoidal Rule and Simpson’s Rule along with error comparison.

Explanation

Divide the interval [a, b] into n sub-intervals of equal width say Δx that is

Δx=ban

Denote each of the intervals as follows:

[x0,x1], [x1,x2], [x2,x3],...,[xn1,xn], where x0=a,xn=b

Midpoint Rule:

For Midpoint Rule, take xi* as the midpoint of each interval that is

x1*=x0+x12,x2*=x1+x22,x3*=x2+x32,...,xn*=xn1+xn2

So, the approximate value of the integral abf(x)dx using Midpoint Rule is

abf(x)dxΔx[f(x1*)+f(x2*)+...+f(xn*)]

Trapezoidal Rule:

The approximate value of the integral abf(x)dx using Trapezoidal Rule is

abf(x)dxΔx2[f(x0)+2{f(x1)+f(x2)+..

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