/*! 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 22 Sketch the region bounded by the... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 11: Problem 22

Sketch the region bounded by the graphs of the functions and find the area of the region. $$ y=\frac{e^{1 / x}}{x^{2}}, y=0, x=1, x=3 $$

Short Answer

Expert verified
The area of the region bounded by the given curves is \(e^{1/3} - e\).

Step by step solution

01

Plot the Functions

It's important to get a visual understanding of the problem. By plotting the given function \(y=\frac{e^{1 / x}}{x^{2}}\) and the lines \(y=0\), \(x=1\), and \(x=3\), we can observe the region whose area we seek. This region is enclosed by the four bounded entities.
02

Setup the Integral

Since we are looking for the area bounded by the curves, we set up an integral from 1 to 3 of the function \(\frac{e^{1 / x}}{x^{2}}\) with respect to \(x\). The integral will be \(\int_1^3 \frac{e^{1 / x}}{x^{2}} dx\).
03

Solve the Integral

We make use of integral calculus to solve the definite integral. Substitution is our best bet in solving this integral. Let's set \(u = 1/x\), then \(du = -1/x^2 dx\) or \( - du = 1/x^2 dx\). Our integral will be \(- \int_{1/3}^{1} e^{u} du\). Now we perform the integration and we get \(-[e^u]_{1/3}^{1}\).
04

Evaluate the Definite Integral

Evaluating the integral we get \(-[e - e^{1/3}]\).
05

Simplify the Result

After simplifying, we get the area between the curves as \(e^{1/3} - e\).

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 graphing utility to graph the region bounded by the graphs of the functions. Write the definite integrals that represent the area of the region. (Hint: Multiple integrals may be necessary.) $$ f(x)=2 x, g(x)=4-2 x, h(x)=0 $$

Find the amount of an annuity with income function \(c(t)\), interest rate \(r\), and term \(T\). $$ c(t)=\$ 250, \quad r=8 \%, \quad T=6 \text { years } $$

Sketch the region bounded by the graphs of the functions and find the area of the region. $$ f(y)=\sqrt{y}, y=9, x=0 $$

Use the Midpoint Rule with \(n=4\) to approximate the area of the region. Compare your result with the exact area obtained with a definite integral. $$ f(y)=4 y-y^{2}, \quad[0,4] $$

Use the Midpoint Rule with \(n=4\) to approximate the area of the region bounded by the graph of \(f\) and the \(x\) -axis over the interval. Compare your result with the exact area. Sketch the region. $$ f(x)=x^{2}+3 \quad[-1,1] $$
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Pure Maths

Read Explanation

Geometry

Read Explanation

Discrete Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Probability and Statistics

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.