Start your trial now! First week only $4.99!*arrow_forward*

BuyFind*launch*

4th Edition

James Stewart

Publisher: Cengage Learning

ISBN: 9781337687805

Chapter 5.1, Problem 22E

**(a)**

To determine

**To find:** The expression for area under the curve

Expert Solution

The area under the curve

**Given:**

The function is

The upper limit is

**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,

**Calculation:**

Find the width of the interval

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).

Find the value of

Substitute 0 for *a* and

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

Substitute

Substitute

Therefore, the expression for the area under the curve is

(b)

To determine

**To evaluate:** The limit

Expert Solution

The value of the limit is

**Calculation:**

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

Rearrange the limit function as shown below.

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

On further simplification,

Therefore, the value of the limit is