# The expression for area under the curve y = x 3 . ### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805 ### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

#### Solutions

Chapter 5.1, Problem 22E

(a)

To determine

## To find: The expression for area under the curve y=x3.

Expert Solution

The area under the curve y=x3 is expressed as limni=1n(in)31n_.

### Explanation of Solution

Given:

The function is y=f(x)=x3.

The upper limit is b=1 and lower limit is a=0.

Formula used:

The area A of the region (S) under the graph f of a continuous function is the sum of area of the approximating rectangles is given by,

A=limni=1nf(xi)Δx (1)

Calculation:

Find the width of the interval (Δx) using the relation:

Δx=ban (2)

Here, the upper limit is b, the lower limit is a, and the number of rectangles is n.

Substitute 1 for b and 0 for a in Equation (2).

Δx=10n=1n

Find the value of xi using the relation:

xi=a+iΔx (3)

Substitute 0 for a and 1n for Δx in Equation (3).

xi=0+i1n=in

Use definition 2 to obtain the expression for area under the curve.

Substitute xi3 for f(xi) and 1n for Δx in equation (1).

A=limni=1nxi31n (4)

Substitute in for xi in equation (4).

A=limni=1n(in)31n

Therefore, the expression for the area under the curve is limni=1n(in)31n_.

(b)

To determine

### To evaluate: The limit limn→∞∑i=1n(in)3⋅1n.

Expert Solution

The value of the limit is 14_.

### Explanation of Solution

Calculation:

The general expression of sum of cubes for the first n integers is shown below:

13+23+33+...+n3=[n(n+1)2]2

Rearrange the limit function as shown below.

limni=1n(in)31n=limni=1ni3n31n=limn1n4i=1ni3 (1)

Apply the general expression of sum of cubes in equation (1).

limni=1n(in)31n=limn1n4[n(n+1)2]2=limn1n4[n2(n+1)24]=14limn[(n+1)2n2]=14limn[n2+2n+1n2]

On further simplification,

limni=1n(in)31n=14limn[1+2n+1n2]=14limn[(1+1n)2]=14(1+1)2=14(1+0)2

=14

Therefore, the value of the limit is 14_.

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