   Chapter 7.7, Problem 19E

Chapter
Section
Textbook Problem

# (a) Find the approximations T 8 and M 8 for the integral ∫ 0 1 cos ( x 2 )   d x .(b) Estimate the errors in the approximations of part (a).(c) How large do we have to choose n so that the approximations T n and M n to the integral in part (a) are accurate to within 0.0001?

To determine

(a)

To find: Integral approximate using Trapezoid rule and midpoint rule

Explanation

Formula used:

Tn=Δx2[f(0)+2i=1n1f(xi)+f(x)]

Mn=Δxi=1nf(x¯i)

Where x¯is are midpoint of subintervals

Given: I=01cos(x2)dx,f(x)=cos(x2)a=0,b=1,n=8

Δx=ban=18=0.125

For T8

T8=Δx2[f(0)+2f(0.125)+2f(0.25)+2f(0.375)+2f(0

To determine

(b) To find: Error estimates in the approximations

To determine

(c) To find:n such that

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