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Chapter 6: Q 58 (page 524)

Use definite integrals to find the volume of each solid of revolution described in Exercises. (It is your choice whether to use disks/washers or shells in these exercises.)

The region between the graph offx=2lnx,y=0,y=3andx=0,revolved around the x-axis.

Short Answer

Expert verified

The required volume by using shells isV鈮�12.5蟺.

Step by step solution

01

Step 1. Given Information

We have given a function :-

fx=2lnx

We have to find the volume of region of graph of this function and y=0,y=3andx=0, revolved around the x-axis.

02

Find the integral and evaluate it to calculate volume 

We know that by using shells the volume is given by :-

V=2蟺鈭�cdr(y)h(y)dy.

Here axis of revolution is x-axis. So ry=yand height is given by the function h(y)=ey2.

Then we get the volume as following :-

V=2蟺鈭�03yey2dy鈬�V=2蟺y22ey2-4ey203鈬�V=2蟺92e32-4e32-0-4鈬�V=2蟺12e32+4鈬�V=2蟺e32+82鈬�V鈮�蟺4.5+8鈬�V鈮�12.5蟺

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Most popular questions from this chapter

Sketching a representative disks ,washers and shells : sketch a representative disks , washers , shells for the solid obtained by revolving the regions shown in figure around the given lines .

The x axis

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 x-axis .

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.

f(x)=9-x2,a,b=-3,3

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

35.dydx=3y,y0=2

Consider the region between the graph of f(x)=x-2 and the x-axis on [2,5]. For each line of rotation given in Exercises 35鈥� 40, use definite integrals to find the volume of the resulting solid.
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