Chapter 5.2, Problem 62E

### Calculus: Early Transcendental Fun...

7th Edition
Ron Larson + 1 other
ISBN: 9781337552516

Chapter
Section

### Calculus: Early Transcendental Fun...

7th Edition
Ron Larson + 1 other
ISBN: 9781337552516
Textbook Problem

# Finding Area by the Limit Definition In Exercises 57-62, use the limit process to find the area of the region bounded by the graph of the function and the y-axis over the given y-interval. Sketch the region. h ( y ) = y 3 + 1 ,             1 ≤ y ≤ 2

To determine

To calculate: The area of the region bounded by the function f(y)=y3+1, in the interval [1,2] and the y-axis.

Explanation

Given: The provided function is,

f(y)=y3+1, in the interval [1,2].

Formula used: The sum of squares of first n natural numbers is given by the formula,

âˆ‘i=1ni2=n(n+1)(2n+1)6

The sum of first n natural numbers is given by the formula,

âˆ‘i=1ni=n(n+1)2

Sum of a constant n times is written as,

âˆ‘i=1nc=nc

The sum of cubes of first n natural numbers is given by the formula,

âˆ‘i=1ni3=n2(n+1)24

Using upper endpoints area is,

A=limnâ†’âˆžâˆ‘i=1nf(Mi)(Î”x)

Here, Mi is the upper endpoints.

Calculation: The provided function is,

f(y)=y3+1

Differentiate f(y) with respect to y and obtain f'(y),

f'(y)=3y2

Equate f'(y) to zero to get,

3y2=0y=0

So, the slope of the function w.r.t y-axis is zero at one point. Check where the curve is approach by put y=âˆž and y=âˆ’âˆž.

On put y=âˆž

f(âˆž)=âˆž3+1

So, f(âˆž) is approach to âˆž.

Now put y=âˆ’âˆž

f(âˆ’âˆž)=âˆ’âˆž3+1

That means the curve is approaching to âˆ’âˆž as yâ†’âˆ’âˆž.

Therefore, the required graph with the region is shown below,

Now, as the provided function f(y) is continuous and non-negative in the interval [1,2]. So, by the partition of interval into n subintervals each of equal width. Therefore,

Consider the width of each rectangle is (Î”y) then,

The width of each rectangle is,

Î”y=2âˆ’1n=1n

The area can be calculated by the use of lower endpoints (mi) or upper endpoints (Mi). For this problem the upper endpoints are convenient.

Now, find upper endpoints (Mi).

Mi=1+in=i+nn

So, the upper endpoint is Mi=i+nn.

Therefore, apply the formula to find the area (A) of the region,

A=limnâ†’âˆžâˆ‘i=1nf(Mi)(Î”y)

Now, substitute the values to find the area of the region

### Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

#### The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started