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Chapter 6: Q. 47 (page 523)

Consider the region betweenf(x)=(x-2)2 and g(x) = x on [1, 4]. For each line of rotation given in Exercises 45鈥�48, use the shell method to construct definite integrals to find the volume of the resulting solid.

Short Answer

Expert verified

The integral can be given as

2蟺鈭�cd(-y2+yy+y+y+2)dy

And the value of the integral is19.13蟺.

Step by step solution

01

Given information

We are given functionsf(x)=(x-2)2 and g(x) = x

also we have,

02

Find the integral and evaluate it

We know that the integral can be given as V=2蟺鈭�cdr(y)h(y)dyas the axis of rotation is y=-1 the radius can be given as r(y)=(y+1)and the height can be given as h(y)=(y+2-y)

Hence the integral can be given as

V=2蟺鈭�cd(y+1)(y+2-y)dyV=2蟺鈭�cd(-y2+yy+y+y+2)dyV=2蟺[-y33+25y52+23y32+y22+2y]41 V=2蟺[12.8-3.23]V=2蟺[9.56]V=19.13蟺

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Most popular questions from this chapter

Sketching disks ,washers and shells : sketch the three disks , washers , shells that result from revolving the rectangles shown in the figure around the given lines

The linex=-1

Use antidifferentiation and/or separation of variables to solve each of the initial-value problems in Exercises 29鈥�52.

32.dydx=-5y,y0=2

Consider the region between f(x)=xand the x-axis on 0,4. For each line of rotation given in Exercises 27鈥�30, use four disks or washers based on the given rectangles to approximate the volume of the resulting solid.

Find the exact value of the arc length of each function f (x) on [a, b] by writing the arc length as a definite integral and then solving that integral .

fx=x32,a,b=0,2

Use antidifferentiation and/or separation of variables to solve each of the initial-value problems in Exercises 29鈥�52

dydx=9.8-0.3150y,y(0)=1000
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