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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

Use 6 subintervals of the same size to approximate the area under the graph of y   3 x 2 from x   =   0 to x   =   1 . Use the right-hand endpoints of the subintervals to find the heights of the rectangles.

To determine

To calculate: The area under the graph of y=3x2 from x=0 to x=1 using the 6 sub-intervals of the same size where the function is evaluated at right-hand end points of each subinterval.

Explanation

Given Information:

The curve is y=3x2 from x=0 to x=1 and the interval [0,1] is divided into 6 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 the 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=3x2 from x=0 to x=1 and the interval [0,1] is divided into 6 subintervals and the function is evaluated at right-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. Also, n=6

Thus,

Base of each rectangle=106=16

Thus, the 6 subintervals, each of length 16, are [0,16], [16,26],[26,36],[36,46],[46,56],[56,1].

Since, there are 6 subintervals, the number of rectangles is 6.

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 [0,16] is 16.

Thus, the height of the first rectangle is,

y=3x2=3(16)2=3(136)=112

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

Thus, the area of the first rectangle is,

Area=base×height=16×112

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

Rectangle Base Right endpoint Height Area=base×height
[0,16] 16 x1=16 y=3(16)2 16×3(16)2
[16,26] 16 x2=26 y=3(26)2 16×3(26)2
[26,36] 16 x3=36 y=3(36)2 16×3(36)2
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Chapter 13 Solutions

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