   Chapter 4, Problem 13PS

Chapter
Section
Textbook Problem

Proof Let f be integrable on [a, b] and 0 < m ≤ f ( x ) ≤ M for all x in the interval [a, b]. Prove that m ( a − b ) ≤ ∫ a b f ( x ) d x ≤ M ( b − a ) . Use this result to estimate ∫ 0 1 1 + x 4 d x .

To determine

To prove: The given statements m(ba)abf(x)dxM(ba)

Explanation

Given:

For any integral function f(x) on [ a,b ] with 0<mf(x)M

m(ba)abf(x)dxM(ba)

Proof:

Consider the properties that for two integral functions f(x),g(x) with f(x)g(x)

abf(x)dxabg(x)dx

Consider f(x)m therefore,

abf(x)dxabmdx

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