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Chapter 6: Q 54 (page 499)

Use definite integrals to find the volume of each solid of revolution described in Exercises 49-61. (It is your choice whether to use disks/washers or shells in these exercises.)

The region bounded the graph of f(x)=exand the line y=eon localid="1651390612178" 0,1,revolved around the x-axis.

Short Answer

Expert verified

The required volume by using shells isV=蟺2

Step by step solution

01

Step 1. Given Information

We have given a function :-

f(x)=ex.

We have to find the volume of region of graph of this function and the liney=eon0,1, 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 that ry=yand the height is hx=lny.

Then we get the volume as following:-

localid="1651817149248" V=2蟺鈭�01ylnydy鈬�V=2蟺y22lny-y2401鈬�V=2蟺12ln1-14-0-0鈬�V=2蟺0-14+0鈬�V=2蟺-14鈬�V=-蟺2

Also, volume cannot be negative so remove the negative sign, then we have:-

localid="1651817155239" V=蟺2

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Most popular questions from this chapter

Suppose your bank account grows at 3percent interest yearly, so that your bank balance after tyears is B(t)=B0(1.03)t.

(a) Show that your bank balance grows at a rate proportional to the amount of the balance.

(b) What is the proportionality constant for the growth rate, and what is the corresponding differential equation for the exponential growth model of B(t)?

Consider the region between the graph of f(x)=x-2and 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.

Use antidifferentiation and/or separation of variables to solve the given differential equations. Your answers will involve unsolved constants.

dydx=xe鈭�y

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

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

dydx=x1+x2
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