/*! 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 29 Let \(R\) be the region bounded ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 29

Let \(R\) be the region bounded by the following curves. Use the washer method to find the volume of the solid generated when \(R\) is revolved about the \(x\) -axis. $$y=e^{x / 2}, y=e^{-x / 2}, x=\ln 2, x=\ln 3$$

Short Answer

Expert verified
Answer: The volume of the solid generated is \(\frac{1}{2}\pi\) cubic units.

Step by step solution

01

Identify the outer and inner radius

Since we are revolving the region around the x-axis, the outer radius in our problem is the distance from the x-axis to the curve \(y=e^{x/2}\), which would simply be the equation itself, so the outer radius is \(r_o(x)=e^{x/2}\). Meanwhile, the inner radius is the distance from the x-axis to the curve \(y=e^{-x/2}\), so the inner radius is \(r_i(x)=e^{-x/2}\).
02

Set up the integral

According to the washer method, the volume of the solid is given by the integral of the difference of the squared outer and inner radii, multiplied by the area of the washer at each x. So, we have the following formula for the volume: $$V = \pi\int_a^b [(r_o(x))^2-(r_i(x))^2]dx$$ In our case, we'll be integrating from \(x=\ln{2}\) to \(x=\ln{3}\): $$V = \pi\int_{\ln{2}}^{\ln{3}} [(e^{x/2})^2-(e^{-x/2})^2]dx$$
03

Simplify the integral

Simplify the expressions inside the integral: $$V = \pi\int_{\ln{2}}^{\ln{3}} [e^x-e^{-x}]dx$$
04

Integrate

Now we integrate the expression with respect to x: $$V = \pi [e^x+e^{-x}]|_{\ln{2}}^{\ln{3}}$$
05

Evaluate the definite integral

We now plug in the limits of integration to obtain the volume: $$V = \pi [(e^{\ln{3}}+e^{-\ln{3}})-(e^{\ln{2}}+e^{-\ln{2}})]$$ Recall that \(e^{\ln{a}}=a\): $$V = \pi [(3+1/3)-(2+1/2)]$$ $$V = \pi [3\frac{1}{3}-2\frac{1}{2}]$$
06

Simplify the expression for the volume

Calculate the result to find the volume of the solid: $$V = \pi [3-\frac{5}{2}]$$ $$V = \frac{1}{2}\pi$$ The volume of the solid generated when the region R is revolved around the x-axis is \(\frac{1}{2}\pi\) cubic units.

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!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Solid of Revolution
When a two-dimensional shape is rotated around an axis, the result is a three-dimensional object known as a solid of revolution. This method is a key concept in calculus to determine the volume of an object whose cross-section is known. Imagine taking a piece of paper with a shaded region and spinning it around a pencil; the shape you create is analogous to the solid of revolution.

Calculus offers a systematic approach to calculate the volume of such objects using the washer or disk method. These methods involve slicing the solid into thin pieces, calculating the volume of each slice, and summing them up鈥攁n application of Riemann sums. The washer method specifically accounts for hollow centers, like a donut shape, hence the name 'washer'.
Definite Integral
The definite integral is crucial for finding the exact volume of solids of revolution. It's a fundamental concept in calculus that quantifies the accumulation of quantities, such as areas under curves and, as in our scenario, volumes. When applied, the definite integral sums up an infinite number of infinitesimally thin washers to find the solid's exact volume.

The process of evaluation involves finding the antiderivative of the function, then applying the Fundamental Theorem of Calculus which suggests subtracting the antiderivative evaluated at the lower limit from that evaluated at the upper limit. The step-by-step solution shows how this principle is applied to find the volume of the solid.
Exponential Functions
Exponential functions play a central role in modeling growth or decay processes, and they have distinctive characteristics such as a constant ratio over equal intervals. In our exercise, we deal with exponential functions of the form y = e^{x/2} and y = e^{-x/2}, where e is the base of the natural logarithm, approximately equal to 2.71828.

These functions grow or decay at a rate proportional to their current value鈥攁 critical feature that often occurs in real-world applications like population growth, radioactive decay, or continuously compounded interest. Understanding their behavior is essential when setting up integrals for calculating volumes of solids of revolution since the radius of the washers often depends on these exponential expressions.

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

Suppose the acceleration of an object moving along a line is given by \(a(t)=-k v(t),\) where \(k\) is a positive constant and \(v\) is the object's velocity. Assume that the initial velocity and position are given by \(v(0)=10\) and \(s(0)=0,\) respectively. a. Use \(a(t)=v^{\prime}(t)\) to find the velocity of the object as a function of time. b. Use \(v(t)=s^{\prime}(t)\) to find the position of the object as a function of time. c. Use the fact that \(d v / d t=(d v / d s)(d s / d t)\) (by the Chain Rule) to find the velocity as a function of position.

Determine whether the following statements are true and give an explanation or counterexample. a. When using the shell method, the axis of the cylindrical shells is parallel to the axis of revolution. b. If a region is revolved about the \(y\) -axis, then the shell method must be used. c. If a region is revolved about the \(x\) -axis, then in principle, it is possible to use the disk/washer method and integrate with respect to \(x\) or the shell method and integrate with respect to \(y\)

Use the following argument to show that \(\lim _{x \rightarrow \infty} \ln x=\infty\) and \(\lim _{x \rightarrow 0^{+}} \ln x=-\infty\). a. Make a sketch of the function \(f(x)=1 / x\) on the interval \([1,2] .\) Explain why the area of the region bounded by \(y=f(x)\) and the \(x\) -axis on [1,2] is \(\ln 2\) b. Construct a rectangle over the interval [1,2] with height \(\frac{1}{2}\) Explain why \(\ln 2>\frac{1}{2}\). c. Show that \(\ln 2^{n}>n / 2\) and \(\ln 2^{-n}<-n / 2\). d. Conclude that \(\lim _{x \rightarrow \infty} \ln x=\infty\) and \(\lim _{x \rightarrow 0^{+}} \ln x=-\infty\).

When an object falling from rest encounters air resistance proportional to the square of its velocity, the distance it falls (in meters) after \(t\) seconds is given by \(d(t)=\frac{m}{k} \ln (\cosh (\sqrt{\frac{k g}{m}} t)),\) where \(m\) is the mass of the object in kilograms, \(g=9.8 \mathrm{m} / \mathrm{s}^{2}\) is the acceleration due to gravity, and \(k\) is a physical constant. a. A \(\mathrm{BASE}\) jumper \((m=75 \mathrm{kg})\) leaps from a tall cliff and performs a ten-second delay (she free-falls for \(10 \mathrm{s}\) and then opens her chute). How far does she fall in \(10 \mathrm{s} ?\) Assume \(k=0.2\) b. How long does it take her to fall the first \(100 \mathrm{m} ?\) The second \(100 \mathrm{m} ?\) What is her average velocity over each of these intervals?

Use a calculator to evaluate each expression or state that the value does not exist. Report answers accurate to four decimal places. a. \(\coth 4\) b. \(\tanh ^{-1} 2\) c. \(\operatorname{csch}^{-1} 5\) d. \(\left.\operatorname{csch} x\right|_{1 / 2} ^{2}\) e. \(\left.\ln \left|\tanh \left(\frac{x}{2}\right)\right|\right|_{1} ^{10}\) f. \(\left.\tan ^{-1}(\sinh x)\right|_{-3} ^{3}\) g. \(\left.\frac{1}{4} \operatorname{coth}^{-1} \frac{x}{4}\right|_{20} ^{36}\)
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Probability and Statistics

Read Explanation

Applied Mathematics

Read Explanation

Calculus

Read Explanation

Statistics

Read Explanation

Discrete Mathematics

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.