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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

Emissions With U.S. Department of Energy data for selected years from 2000 and projected to 2030, sulphur dioxide emissions from electricity generation (in millions of short tons per year) can be modeled by

E (   x ) =   0.0112 x 2 + 0.612 x + 11.9 where x is the number of years past 2000. Use n = 10 equal subdivisions and right-hand endpoints to approximate (to the nearest unit) the area under the graph of E(x) between x = 15 and x = 25. What does this area represent?

To determine

To calculate: The area under the graph of E(x)=0.0112x2+0.612x+11.9 between x=15 and x=25 using n=10 equal subdivisions measured at the right-handend points and what does this area represent where the function represents sulphur dioxide emissions from electricity generation.

Explanation

Given Information:

The provided function is E(x)=0.0112x2+0.612x+11.9.

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.

Calculation:

Consider the function,

E(x)=0.0112x2+0.612x+11.9

The range of x-axis is from x=15 to x=25.

Divide the interval [15,25] into n=10 subintervals where the function is evaluated at right 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=15 to x=25.So, a=15 and b=25.

Thus,

Base of each rectangle=251510=1010=1

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.

Record these values in a table,

Rectangle Base Right endpoint Height Area=base×height
[15,16] 1 x1=16 E(16)=0.0112(16)2+0.612(16)+11.9=24.5592 1×24.5592=24.5592
[16,17] 1 x2=17 E(17)=0.0112(17)2+0.612(17)+11.9=25.5408 1×25.5408=25.5408
[17,18] 1 x3=18 E(18)=0.0112(18)2+0.612(18)+11.9=26.5448 1×26.5448=26.5448
[18,19] 1 x4=19 E(19)=0.0112(19)2+0.612(19)+11

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