   Chapter 4.1, Problem 26E

Chapter
Section
Textbook Problem

# (a) Use Definition 2 to find an expression for the area under the curve y =   x 3 from 0 to 1 as a limit.(b) The following formula for the sum of the cubes of the first n integers is proved in Appendix E. Use it to evaluate the limit in part (a). 1 3 + 2 3 + 3 3 + ... + n 3 = [ n ( n − 1 ) 2 ] 2

To determine

a)

To find:

An expression for the area under the curve

Explanation

1) Concept:

The area A of the region S that lies under the graph of the continuous function f is

A=limni=1nfxix

x=b-an, xi=a+ix

2) Given:

y=x3,  0x1

3) Calculation:

It is given that the curve

y=x3,  0x1

Here a=0,   b=1

x=b-an=1-0n=1n

x=1n

xi=a+ix

=0+i·1n

=in

xi=in

Therefore, the a

To determine

b)

To evaluate:

limni=1ni3n4

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