   Chapter 5.5, Problem 90E ### 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

# If f is continuous on ℝ , prove that ∫ a b f ( x + c )   d x = ∫ a + c b + c f ( x )   d x For the case where f(x) ≥ 0, draw a diagram to interpret this equation geometrically as an equality of areas.

To determine

To prove: The integral function as abf(x+c)dx=a+cb+cf(x)dx.

Explanation

Given:

The integral function of left hand side (LHS) is abf(x+c)dx.

The region lies between x=a and x=b.

Calculation:

Consider u=x+c (1)

Differentiate both sides of the Equation (1).

du=dx

Calculate the lower limit value of u using Equation (1).

Substitute a for x in Equation (1).

u=a+c

Calculate the upper limit value of u using Equation (1).

Substitute b for x in Equation (1).

u=b+c

The integral function is,

abf(x+c)dx (2)

Apply lower and upper limits for u in Equation (2).

Substitute u for (x+c) and du for dx in Equation (2).

abf(x+c)dx=a+cb+cf(u)du (3)

Consider u=x (4)

Differentiate both sides of the Equation (4)

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