   Chapter 5.2, Problem 21E ### 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

# Use the form of the definition of the integral given in Theorem 4 to evaluate the integral. ∫ 2 5 ( 4 − 2 x ) d x

To determine

To evaluate: The integral function 25(42x)dx using Theorem 4.

Explanation

Given:

The integral function is 25(42x)dx, where the interval [a,b] is [2,5].

Definition used:

Apply theorem 4.

abf(x)dx=limni=1nf(xi)Δx, where, f(x) is continuous on [a,b] and Δx width of interval.

Calculation:

Calculate Δx using the relation Δx=ban as,

Δx=ban=52n=3n

Calculate xi using the relation xi=a+iΔx as,

xi=a+iΔx=2+3in

Consider the function f(x)=(42x), from the given integral 25(42x)dx of the form abf(x)dx.

Substitute xi for x and the function f(x)=(42x) becomes,

f(xi)=(42xi)

Calculate f(xi) using Equation (2).

Substitute (2+3in) for xi in f(xi)=(42xi),

f(xi)=(42(2+3in))=(446in)=(6in)

Substitute the above values in abf(x)dx=limn

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