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Chapter 6: Q. 31 (page 523)

Consider the region between f(x)=4鈭�x2and the x-axis on [0,2]. For each line of rotation given in Exericses 29鈥�32, use the shell method to construct definite integrals to find the volume of the resulting solid.

Short Answer

Expert verified

The volume is 403蟺.

Step by step solution

01

Step 1. Given Information. 

We are given,

02

Step 2. Finding the Volume. 

As the region bounded by f(x)=4-x2 and the x-axis from x=0, to x=2is rotated around the line x=2, so to find the volume by shell method note that the radius will be 2-xand the height of the shell is given by y=f(x)=4-x2

Therefore using the shell method

Volume=2蟺鈭�02(2-x)4-x2dx=2蟺鈭�02x3-2x2-4x+8dx=2蟺14x4-23x3-2x2+8x02=2蟺4-163-8+16-[0]=403蟺

Hence, the volume is 403蟺.

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