Chapter 6: Q. 31 (page 523)

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

Short Answer

Expert verified

The volume is $\frac{40}{3} π$ .

Step by step solution

Step 1. Given Information.

We are given,

Step 2. Finding the Volume.

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

Therefore using the shell method

$Volume = 2 π \int_{0}^{2} (2 - x) (4 - x^{2}) d x = 2 π \int_{0}^{2} (x^{3} - 2 x^{2} - 4 x + 8) d x = 2 π {[\frac{1}{4} x^{4} - \frac{2}{3} x^{3} - 2 x^{2} + 8 x]}_{0}^{2} = 2 π ([4 - \frac{16}{3} - 8 + 16] - [0]) = \frac{40}{3} π$

Hence, the volume is $\frac{40}{3} π$ .

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

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Finding the Volume.

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

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Finding the Volume.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Discrete Mathematics

Pure Maths

Probability and Statistics

Mechanics Maths

Statistics

Study anywhere. Anytime. Across all devices.

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