   Chapter 4.5, Problem 61E

Chapter
Section
Textbook Problem

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

To determine

To prove:

abf(-x)dx=-b-af(x)dx for fx0and 0<a<b

Also draw a diagram to interpret this equation geometrically as an equality of areas

Explanation

1) Concept:

i. The substitution rule: If u=g(x) is a differentiable function whose range is I and f is continuous on I, then

f(gx)g'xdx=f(u)du.

Here g(x) is substituted as u and then g(x)dx =du

ii.

abf(x)dx=-baf(x)dx

iii.

abf(x)dx=abf(u)du

2) Given:

f  is continuous on R

3) calculation:

Let

abf(-x)dx

Here use the substitution method

Substitute -x=u

Differentiating with respect to x

-dx=du

dx=-du

The limits changes the new limits of integration calculated by substituting

At x=a, u=-a and

At x=b, u=-b

Therefore, the integral becomes from -a to-b

abf(-x)dx=-a-b-f(u)

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