   Chapter 5.2, Problem 68E ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

#### Solutions

Chapter
Section ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

# Which of the integrals ∫ 0 0.5 cos ( x 2 ) d x , ∫ 0 0.5 cos x   d x is larger? Why?

To determine

To find: The larger value of integral among the integral function 00.5cos(x2)dx and 00.5cosxdx.

To explain: The reason for the larger value of the integral.

Explanation

Consider the Cosine function is a decreasing function within limits [0,0.5] and the value of x2 is lesser than x within limits [0,0.5]. Thus, the value of 00.5cos(x2)dx is higher than 00.5cosxdx.

Given information:

The integral function 00.5cos(x2)dx and 00.5cosxdx within limits [0,0.5].

Calculation:

Consider the value of function as follows:

f(x)=cosx2 (1)

g(x)=cosx (2)

Calculate the value of f(x) within limits [0,0.5] using Equation (1).

Calculate the value of f(0) using Equation (1).

Substitute 0 for x in Equation (1).

f(0)=cos(0)=1

Similarly, calculate for various x values and tabulate the values as in table 1.

 x f(x)=cosx2 0 1 0.1 0.99995 0.2 0.9992 0.3 0.9959 0.4 0.9872 0.5 0.9689

Table 1

Refer Table 1,

The function f(x)=cosx2 is a decreasing function within limits [0,0.5].

Calculate the value of g(x) within limits [0,0

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Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th 