   Chapter 13.1, Problem 29E ### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042

#### Solutions

Chapter
Section ### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042
Textbook Problem

# For parts (a)-(e), use the function y   =   x 2  from  x =   0  to  x =   1  with  n equal subintervals and the function evaluated at the right-hand endpoints.(a) Find a formula for the sum of the areas of the n rectangles. (Call this S.) Then find ( b )       S ( 10 ) .                   ( c )   S ( 100 ) . ( d )   S ( 1000 ) .                     ( e ) lim n → ∞ S .

(a)

To determine

To calculate: A formula for the sum of the areas of the n rectangles under the curve of the function y=x2 from x=0 to x=1 where the function is evaluated at right-hand end points of each subinterval.

Explanation

Given Information:

The curve is y=x2 from x=0 to x=1 and the interval [0,1] is divided into n subintervals and the function is evaluated at right-hand end points of the subintervals.

Formula used:

The base of the rectangles to approximate the area is ban where the interval [a,b] on which function is defined is divided into n subintervals.

The height of the rectangles is the value of the function calculated at the right-hand end point of the interval containing the base.

The area of a rectangle is base×height.

The approximated area under the curve is sum of the areas of each rectangle.

The value of the sum k=1nk2=n(n+1)(2n+1)6.

Calculation:

The curve is y=x2 from x=0 to x=1 and the interval [0,1] is divided into n subintervals and the function is evaluated at left-hand end points of the subintervals.

The base of the rectangles to approximate the area is ban where the interval [a,b] on which function is defined is divided into n subintervals.

Since, the function is defined from x=0 to x=1. So, a=0 and b=1.

Thus,

Base of each rectangle=10n=1n

Thus, the n subintervals, each of length 1n, are [0,1n], [1n,2n],,[n1n,1].

Since, there are n subintervals, the number of rectangles is n.

Recall that the height of the rectangles is the value of the function calculated at the right-hand end point of the interval containing the base.

Since, the right-hand end point of the first subinterval [0,1n] is 1n.

Thus, the height of the first rectangle is,

y=x2=(1n)2=1n2

Recall that the area of a rectangle is base×height.

Thus, the area of the first rectangle is,

Area=base×height=1n×1n2=1n3

Do a similar calculation to find the area of all n rectangles and record these values in a table

(b)

To determine

To calculate: The value of S(10) using the formula S=(n+1)(2n+1)6n2 for the sum of the areas of the n rectangles.

(c)

To determine

To calculate: The value of S(100) using the formula S=(n+1)(2n+1)6n2 for the sum of the areas of the n rectangles.

(d)

To determine

To calculate: The value of S(1000) using the formula S=(n+1)(2n+1)6n2 for the sum of the areas of the n rectangles.

(e)

To determine

To calculate: The value of limnS where S=(n+1)(2n+1)6n2 is the formula for the sum of the areas of the n rectangles under the curve of the function y=x2 from x=0 to x=1.

### 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 