   Chapter 4.2, Problem 70E

Chapter
Section
Textbook Problem

# (a) If f is continuous on [a, b], show that | ∫ a b f ( x ) d x | ≤ ∫ a b | f ( x ) | d x [Hint: − | f ( x ) | ≤ f ( x ) ≤ | f ( x ) | .](b) Use the result of part (a) to show that | ∫ 0 2 π f ( x ) sin 2 x d x | ≤ ∫ 0 2 π | f ( x ) | d x

To determine

a)

To show:

abfxdxabfxdx

Explanation

1) Concept:

i. Theorem:If f  is continuous on a, b, or if f  has only a finite number of jump discontinuities, then f  is integrable on a, b

ii. Comparison property of the integral

If fxgx for a x b , then abfxdx abgxdx

2) Calculation:

As the value of a function at a point lies in between its negative absolute value and positive absolute value we have

-fxfxfx

As f  is continuous on a, b,  it is integrable on a, b

So, integrate from a, b

ab-fxdxabfxdx

To determine

b)

To show:

02πfxsin2xdx02πfxdx

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