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Chapter 6: Q. 49 (page 523)

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

The region between the graph of f(x)=4-x2and the line y = 4 on [0, 2], revolved around the y-axis.

Short Answer

Expert verified

The value of the volume is 8蟺cubic units.

Step by step solution

01

Given information

We are given a functionf(x)=4-x2and y = 4

02

Find the integral and evaluate it

We know that the volume can be given as V=2蟺鈭�cdr(x)h(x)dx. The axis of revolution is y-axis

Hence the radius can be given as r(x)=xand the height is h(x)=4-x2. Substituting the values in the integral we get,

V=2蟺鈭�02x(4-x2)dxV=2蟺鈭�024x-x3dxV=2蟺[2x2-x44]20 V=2蟺[4]V=8蟺

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Most popular questions from this chapter

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=4-x ,a,b=2,5

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.

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.

32.dydx=-5y,y0=2

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

dydx=4y,y(1)=1
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