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Precalculus: Mathematics for Calcu...

7th Edition
James Stewart + 2 others
Publisher: Brooks Cole
ISBN: 9781305071759

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BuyFindarrow_forward

Precalculus: Mathematics for Calcu...

7th Edition
James Stewart + 2 others
Publisher: Brooks Cole
ISBN: 9781305071759
Chapter 13.5, Problem 8E
Textbook Problem
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Estimating Areas Using Rectangles Approximate the area of the shaded region under the graph of the given function by using the indicated rectangles. (The rectangles have equal width.)

8. f(x) = 9xx3

Chapter 13.5, Problem 8E, Estimating Areas Using Rectangles Approximate the area of the shaded region under the graph of the

To determine

To Find: The approximate area under the function 9xx3 by using rectangles.

Explanation of Solution

Given:

The given function is, f(x)=9xx3 .

Calculation:

Divide the area under the given function into four small rectangles as shown in Figure (1) below.

Figure (1)

The total area under the given function is sum of areas of individual rectangles.

The total area under the function is,

A=f(x1)Δx1+f(x2)Δx2+f(x3)Δx3+f(x4)Δx4+...+f(xn) (1)

Use summation and write equation (1) as,

A=r=1nf(xr)Δx

Here, f(x) is the height of the rectangle, Δx is the width of rectangle.

The width of each rectangle is given by, Δx=ban

Here, b and a are end points of required area, n is number of rectangles.

The end points a and b as shown in Figure (1) are 1 and 1 respectively.

Calculate width of each rectangle, substitute 8 for n, 2 for b and 0 for a in Δx .

Δx=208=0.25

Calculate height f(x1) of first rectangle starting from left of the function.

f(x1)=9(x1)(x1)3=9(0.25)                      [x1=a+Δx]=2.25

Height of second rectangle f(x2)

f(x2)=9(x2)(x2)3=9(0.5)(0.5)3                    [x2=x1+Δx]=4.62

Height of third rectangle f(x3) .

f(x3)=9(x3)(x3)3=9(0.75)(0.75)3                     [x3=x2+Δx]=6

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Chapter 13 Solutions

Precalculus: Mathematics for Calculus (Standalone Book)
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