Q. 18 Let 0<a<b Use the shell me... [FREE SOLUTION]

Chapter 13: Q. 18 (page 1027)

Let $0 < a < b$ Use the shell method to find an integral that represents the volume of the solid of revolution obtained when the region bounded above by the graph of $f$ and bounded below by the $x$ -axis on the interval $[a, b]$ is rotated about the $z$ -axis.

Short Answer

Expert verified

The volume of the solid generated is $V = {鈭�}_{r = a}^{r - b} 2 蟺 r \cos^{2} 胃 f (r) d r$

Step by step solution

Given Information

The objective of this problem is to use the technique of polar coordinates to represent the volume of the solid bounded above by the graph of function $g$ and below by the $x - y$ plane over annulus $a^{2} 鈮� x^{2} + y^{2} 鈮� b^{2}$

Calculation

Use shell method for the volume generated by the region bounded by the graph.

$V = {鈭�}_{0}^{b} 2 蟺 x z d x$

$V = {鈭�}_{a}^{b} 2 蟺 x g (x, y) d x$

$V = {鈭�}_{0}^{b} 2 蟺 x f (\sqrt{x^{2} + y^{2}}) d x$

Substitute $x = r \cos 胃, y = r \sin 胃$

$V = {鈭�}_{r m a}^{r b b} 2 蟺 r \cos 胃 f (\sqrt{r^{2} \cos^{2} 胃 + r^{2} \sin^{2} 胃}) \cos 胃 d r$

$V = {鈭�}_{r - 胃}^{r - t} 2 蟺 r \cos^{2} 胃 f (r) d r$

Thus, the volume of the solid generated is

$V = {鈭�}_{r = a}^{r - b} 2 蟺 r \cos^{2} 胃 f (r) d r$

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Let $0 < a < b$ Use the shell method to find an integral that represents the volume of the solid of revolution obtained when the region bounded above by the graph of $f$ and bounded below by the $x$ -axis on the interval $[a, b]$ is rotated about the $z$ -axis.

Short Answer

Step by step solution

Given Information

Calculation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Pure Maths

Geometry

Decision Maths

Probability and Statistics

Calculus

Study anywhere. Anytime. Across all devices.

91影视

Let 0<a<b Use the shell method to find an integral that represents the volume of the solid of revolution obtained when the region bounded above by the graph off and bounded below by the x-axis on the interval [a,b] is rotated about the z-axis.

Short Answer

Step by step solution

Given Information

Calculation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Pure Maths

Geometry

Decision Maths

Probability and Statistics

Calculus

Study anywhere. Anytime. Across all devices.

Let $0 < a < b$ Use the shell method to find an integral that represents the volume of the solid of revolution obtained when the region bounded above by the graph of $f$ and bounded below by the $x$ -axis on the interval $[a, b]$ is rotated about the $z$ -axis.