   Chapter 4.2, Problem 28E

Chapter
Section
Textbook Problem

# Prove that ∫ a b x 2   d x = b 3 − a 3 3

To determine

To prove:

abx2dx=b3-a33

Explanation

1) Concept:

Use theorem (4) to prove the given statement.

Theorem (4):

If f is integrable on [a, b] then

abfxdx=limni=1nfxi x

where x= b-an and xi=a+i x

2) Formula:

i)i=1ni= nn+12ii)i=1ni2= nn+1(2n+1)6iii)i=1ncai=ci=1naiwhere c is a constantiv)i=1n(ai-bi)=i=1nai-i=1nbi

3) Given:To prove

abx2dx=b3-a33

4) Calculation:

According to theorem (4),

abfxdx=limni=1nfxi x

Substituting the value of xi,

abfxdx=limni=1nfa+i x x

By using formula (iii),

limni=1nfa+i x x=limnxi=1nfa+i x

Here,fx=x2.

Therefore,

abx2 dx=limnxi=1na+i x2

By using (a+b)2=a2+2ab+b2

=limnxi=1n[a2+2a (i x)+i x2]

By using formula (iv),

=limnxi=1na2+i=1n2a*i x+i=1ni2*x2

By using formula (i), (ii) and (iii),

=limnx*a2n+2a*x*x*nn+12+x3* nn+1(2n+1)6

Simplifying,

=limnx

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