Chapter 6: Q. 41 (page 523)

Consider the region between $f (x) = 4 - x^{2}$ and the line y = 5 on [0, 2]. For each line of rotation given in Exercises 39–42, use the shell method to construct definite integrals to find the volume of the resulting solid.

Short Answer

Expert verified

The integral can be given as $2 π \int_{0}^{2} (3 x^{2} + 3 - x^{3} - x) d x$ and its value is $16 π$

Step by step solution

Given information

We are given $f (x) = 4 - x^{2}$ and

Find the integral and evaluate it

We know that integral can be given as

$V = 2 π \int_{0}^{2} r (x) h (x) d x$ where r and h are the radius and height respectively

the axis of revolution is x=3 hence the radius is $r (x) = 3 - x$ and height can be given as $h (x) = x^{2} + 1$ and interval is [0,2]

Substituting the values we get

role="math" localid="1650609717077" $V = 2 π \int_{0}^{2} (3 - x) (x^{2} + 1) d x V = 2 π \int_{0}^{2} (3 x^{2} + 3 - x^{3} - x) d x V = 2 π {[x^{3} + 3 x - \frac{x^{4}}{4} - \frac{x^{2}}{2}]}^{2}_{0} V = 16 π c u b i c u n i t s$

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Consider the region between $f (x) = 4 - x^{2}$ and the line y = 5 on [0, 2]. For each line of rotation given in Exercises 39–42, use the shell method to construct definite integrals to find the volume of the resulting solid.

Short Answer

Step by step solution

Given information

Find the integral and evaluate it

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Consider the region between f(x)=4-x2and the line y = 5 on [0, 2]. For each line of rotation given in Exercises 39–42, use the shell method to construct definite integrals to find the volume of the resulting solid.

Short Answer

Step by step solution

Given information

Find the integral and evaluate it

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Calculus

Theoretical and Mathematical Physics

Decision Maths

Mechanics Maths

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

Consider the region between $f (x) = 4 - x^{2}$ and the line y = 5 on [0, 2]. For each line of rotation given in Exercises 39–42, use the shell method to construct definite integrals to find the volume of the resulting solid.