Chapter 5.3, Problem 3E

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

Evaluating a Limit In Exercises 3 and 4, use Example 1 as a model to evaluate the limit lim n → ∞ ∑ i = 1 n f ( c i ) Δ x i Over the region bounded by the graphs of the equations. f ( x ) = x ,   y = 0 ,   x = 0 ,   x = 3   ( Hint :   Let   c i = 3 i 2 n 2 . )

To determine

To calculate: The Riemann’s limit over the region bounded by the graphs of the equations f(x)=x y=0,x=0,x=3.

Explanation

Given: The provided equations are:

f(x)=x, y=0,â€‰â€‰â€‰x=0,â€‰â€‰â€‰x=3

The limit to evaluate is:

limnâ†’âˆžâˆ‘i=1nf(ci)Î”xi

Formula used:

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

âˆ‘i=1ni=n(n+1)2

The sum of the squares of first n natural numbers is:

âˆ‘i=1ni2=n(n+1)(2n+1)6

Calculation:

The limit to evaluate is

limnâ†’âˆžâˆ‘i=1nf(ci)Î”xi

Here ci is the right endpoints of the partition and Î”xi is the width of the ith interval.

Assume that ci=3i2n2.

Therefore, the width of the ith interval is given as:

Î”xi=3i2n2âˆ’3(iâˆ’1)2n2=3i2âˆ’3(i2+1âˆ’2i)n2=6iâˆ’3n2

So, the limit is:

limnâ†’âˆžâˆ‘i=1nf(ci)Î”xi=limnâ†’âˆžâˆ‘i=1nf(3i2n2)Î”xi

Put value of f(3i2n2) and Î”xi in above equation:

limnâ†’âˆžâˆ‘i=1nf(3i2n2)Î”xi=limnâ†’âˆžâˆ‘i=1n3i2n2(6iâˆ’3n2)=limnâ†’âˆžâˆ‘i=1n3ni(6iâˆ’3n2)

Factor out 3

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

Evaluate 01(x+1x2)dx by interpreting it in terms of areas.

Single Variable Calculus: Early Transcendentals, Volume I

In problems 37-44, perform the indicated operations and simplify. 42.

Mathematical Applications for the Management, Life, and Social Sciences

The general solution to (for x, y > 0) is: a) y = ln x + C b) c) y = ln(ln x + C) d)

Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th