   Chapter 4.2, Problem 24E

Chapter
Section
Textbook Problem

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

To determine

To evaluate:

The integral 022x-x3 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 the 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=1ni= n(n + 1)2

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:

022x-x3 dx

4) Calculation:

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

a=0, b=2 and fx=2x-x3

Substituting the value of a and b in x,

x= b-an

x= 2n

Now find xi.

xi=a+i x

xi=0+i 2n

xi=2in

By using the theorem (4),

022x-x3 dx= limni=1nfxi x

Now, substitute the values of x and xi from above

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