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Chapter 6: Q. 48 (page 523)

Consider the region between f(x)=(x-2)2and 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�14(-x3+4x2+x-4)dx

and the value of integral is31.5蟺cubicunits

Step by step solution

01

Given information

We are given functionsf(x)=(x-2)2andg(x)=x

also

02

Find the integral and evaluate it

We know that integral can be given as V=2蟺鈭�abr(x)h(x)dxthe axis of rotation is r(x)=x+1and the height can be given as h(x)=x-(x-2)2

Substituting the values in the integral we get,

role="math" localid="1650687932200" V=2蟺鈭�14(x+1)(x-(x-2)2)dxV=2蟺鈭�14(-x3+4x2+x-4)dxV=2蟺[-x24+43x3+x22-4x]41V=2蟺[15.75]V=31.5蟺

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