/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 21 Let \(a, b \in \mathbb{R}\) with... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 21

Let \(a, b \in \mathbb{R}\) with \(0 \leq a

Short Answer

Expert verified
The volume of the solid body obtained by revolving the region $R$ about the $y$-axis is equal to: $$ 2 \pi \int_{a}^{b} x 鈩�(x) d x $$

Step by step solution

01

Visualize the problem

Imagine a small line segment in the region R at distance x from the y-axis and having length 鈩�(x). When we revolve this small line segment around the y-axis, it forms a thin cylindrical strip. We can consider the volume of this thin strip and then sum up the volumes of all such thin strips to find the volume of the solid body formed by revolving the region.
02

Calculate the volume of one thin cylindrical strip

Consider a thin strip of width dx at distance x from the y-axis, having length 鈩�(x). When it revolves around the y-axis, it traces out a thin cylindrical surface. The circumference of this circular strip is equal to \(2 \pi x\), and its width is dx. So, the area of the cylindrical surface is \(2 \pi x dx\). Since the length of the line segment is 鈩�(x), the volume of this thin cylindrical strip is given by the product of the area of the cylindrical surface and the length of the line segment intersecting it: $$ dV = 2 \pi x dx \times 鈩�(x) $$
03

Sum up the volumes of all thin strips using integration

To find the volume of the solid body formed by revolving the region, we need to sum up the volumes of all such thin cylindrical strips. This can be done using integration over the interval [a, b]. $$ V = \int_{a}^{b} dV $$ Now substitute the expression for dV from step 2: $$ V = \int_{a}^{b} 2 \pi x 鈩�(x) dx $$ So, the volume of the solid formed by revolving the region R around the y-axis is $$ 2 \pi \int_{a}^{b} x 鈩�(x) d x $$

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

Most popular questions from this chapter

Use a result of Pappus to find (i) the volume of a cylinder with height \(h\) and radius \(a\) (ii) the volume of a cone with height \(h\) and base radius \(a .\)

Let \(a \in \mathbb{R}\) with \(a>0\). The base of a certain solid body is the disk given by \(x^{2}+y^{2} \leq a^{2} .\) Each of its slices by a plane perpendicular to the \(x\) -axis is an isosceles right-angled triangular region with one of the two equal sides in the base of the solid body. Find the volume of the solid body.

Find the volume of the solid generated by revolving the region bounded by the curves given by \(y=\sqrt{x}, y=2\), and \(x=0\) about the \(x\) -axis by both the Washer Method and the Shell Method. If the region is revolved about the line given by \(x=4\), what is the volume of the solid so generated?

Consider the function \(f:[0,1] \rightarrow \mathbb{R}\) defined by \(f(x)=x e^{-x^{2}}\). Find \(T_{n}(f)\) and \(S_{n}(f)\) with \(n=2\) and \(n=4\). Obtain the corresponding error estimates, and compare them with the actual errors $$ \int_{0}^{1} f(x) d x-T_{n}(f) \quad \text { and } \quad \int_{0}^{1} f(x) d x-S_{n}(f) . $$

For each of the following curves, find the arc length as well as the area of the surface generated by revolving the curve about the line given by \(y=-1\) (i) \(y=\frac{x^{3}}{3}+\frac{1}{4 x}, 1 \leq x \leq 3\), (ii) \(x=\frac{3}{5} y^{5 / 3}-\frac{3}{4} y^{1 / 3}, 1 \leq y \leq 8\).
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Geometry

Read Explanation

Mechanics Maths

Read Explanation

Discrete Mathematics

Read Explanation

Logic and Functions

Read Explanation

Theoretical and Mathematical Physics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.