   Chapter 4.2, Problem 25E

Chapter
Section
Textbook Problem

# Use the form of the definition of the integral given in Theorem 4 to evaluate the integral. ∫ 0 1 ( x 3 − 3 x 2 )   d x

To determine

To evaluate:

The integral 01x3-3x2 dx by using the form of the definition of the integral given in the theorem (4).

Explanation

1) Concept:

Use the form of the definition of the integral given in theorem (4).

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=1ni2= nn+1(2n+1)6

ii)i=1ni3= n(n+1)22

iii)i=1ncai=ci=1naiwhere c is a constant

iv)i=1n(ai-bi)=i=1nai-i=1nbi

3) Given:

01x3-3x2 dx

4) Calculation:

Compare the given integral with the theorem (4) that gives

a=0, b=1, and fx=x3-3x2

Substituting value of a and b in x,

x= b-an

x= 1n

Now find xi.

xi=a+i x

xi=0+i 1n

xi=in

By using the theorem (4),

01x3-3x2 dx= limni=1nfxi x

Now, substitute the values of x and xi from the above.

01x3-3x2 dx=limni=1nfin1n

By using the formula (iii),

= limn1ni=1nfin

Therefore,

01x3-3x2 dx= limn1ni=1ni3n3 -3i2n2

By using theformula (iii) and (iv),

=

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