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Mathematical Applications for the ...

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

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BuyFindarrow_forward

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.

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Sect-13.1 P-9ESect-13.1 P-10ESect-13.1 P-11ESect-13.1 P-12ESect-13.1 P-13ESect-13.1 P-14ESect-13.1 P-15ESect-13.1 P-16ESect-13.1 P-17ESect-13.1 P-18ESect-13.1 P-19ESect-13.1 P-20ESect-13.1 P-21ESect-13.1 P-22ESect-13.1 P-23ESect-13.1 P-24ESect-13.1 P-25ESect-13.1 P-26ESect-13.1 P-27ESect-13.1 P-28ESect-13.1 P-29ESect-13.1 P-30ESect-13.1 P-31ESect-13.1 P-32ESect-13.1 P-33ESect-13.1 P-34ESect-13.1 P-35ESect-13.1 P-36ESect-13.1 P-37ESect-13.1 P-38ESect-13.1 P-39ESect-13.1 P-40ESect-13.2 P-1CPSect-13.2 P-2CPSect-13.2 P-1ESect-13.2 P-2ESect-13.2 P-3ESect-13.2 P-4ESect-13.2 P-5ESect-13.2 P-6ESect-13.2 P-9ESect-13.2 P-10ESect-13.2 P-11ESect-13.2 P-12ESect-13.2 P-7ESect-13.2 P-8ESect-13.2 P-23ESect-13.2 P-24ESect-13.2 P-13ESect-13.2 P-14ESect-13.2 P-15ESect-13.2 P-16ESect-13.2 P-17ESect-13.2 P-18ESect-13.2 P-19ESect-13.2 P-20ESect-13.2 P-21ESect-13.2 P-22ESect-13.2 P-25ESect-13.2 P-26ESect-13.2 P-27ESect-13.2 P-28ESect-13.2 P-29ESect-13.2 P-30ESect-13.2 P-31ESect-13.2 P-32ESect-13.2 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