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Chapter 6: Q. 57 (page 512)

Consider the region between the graph of f(x) = 1 鈭� cos x and the x-axis on [0,蟺]. For each line of rotation given in Exercises 55鈥�58, write down definite integrals that represent the volume of the resulting solid and then use a calculator or computer to approximate the integrals.

Short Answer

Expert verified

The volume of the resulting solid is V=18.440.

Step by step solution

01

Step 1. Given

The graph of f(x) = 1 鈭� cos x and the x-axis on [0,蟺].

02

Step 2.  Calculation

To determine the volume of solid of revolution, rotated around vertical line, express the curve as inverse function.

f(x)=1-cosxy=1-cosxcosx=1-yx=cos-1(1-y)p(y)=cos-1(1-y)

Use the definition of function to determine the y-interval

For the x-interval of [0.] the corresponding interval of y-variable will be [0.2]

For the disk the radius is 蟺-p(y)

The volume of a disk is given by the integral

V=蟺鈭�abR(y)2dy

Use this definition and the values determined above to determine the volume of solid of revolution

V=蟺(蟺2-4)V=18.440

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