/*! 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 68 Find the area of the region. $... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 68

Find the area of the region. $$ y=\frac{e^{x}}{1+e^{2 x}} $$

Short Answer

Expert verified
The area of the region defined by the curve \( y = \frac {e^{x}} {1+e^{2x}} \) from \( x = a \) to \( x = b \) is \( ln |1 + e^{b}| - ln |1 + e^{a}| = ln \left| \frac {1 + e^{b}} {1 + e^{a}} \right| \)

Step by step solution

01

Rearrange the Formula

Rearrange the given equation for integration. Here, you can indicate the y as \( dy \) and the other parts of the equation as \( dx \). So the equation becomes \( dy = \frac {e^{x}}{1+e^{2x}} dx \)
02

Integration

Next, integrate both sides of the equation over their respective variables, from a to b (the bounds of the region). Remember that the bounds apply both to y and to the corresponding x-values in the integral. The expression becomes: \( y = \int_a^b \frac {e^{x}}{1+e^{2x}} dx \) where \( a \leq y \leq b \)
03

Solve the Integral

Solving this integral could be a bit tricky. You need to use a special function known as the logistic function. The result is \( y = ln|1 + e^{x}| \) evaluated from a to b.
04

Evaluate from a to b

Finally, you will need to evaluate \( y \) from a to b. This will give you the value of the area. Be sure to handle the absolute value correctly while evaluating the limits.

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.

Definite Integral
Understanding the definite integral is crucial for solving many problems in calculus. It represents the accumulation of quantities, which can be seen as an area under a curve between two points. When you see an expression like \[ \int_a^b f(x) dx \], think of it as the total \(y\)-value accumulated between \(x = a\) and \(x = b\). This concept is essential when calculating the area of a region bounded by a curve and the x-axis within certain limits. For the given problem, we're interested in the region under the curve \(y = \frac{e^{x}}{1+e^{2x}}\) from \(x = a\) to \(x = b\), which is found by integrating that function across those bounds.
Logistic Function
The logistic function is a sigmoid-shaped curve often used to describe growth that starts exponentially, but then tapers off as it approaches a maximum limit. In our study, it helped to solve the integral \[ \int \frac {e^{x}}{1+e^{2x}} dx \]. To understand logistic growth in a more general sense, we use the formula \( L/(1+e^{-k(x-x_0)}) \), where \(L\) is the carrying capacity, \(k\) is the growth rate, and \(x_0\) is the midpoint of the sigmoid curve. When integrating functions that resemble part of the logistic growth curve, they can often be simplified into a form that's more straightforward to evaluate.
Exponential Function
Exponential functions, like \(e^{x}\), describe processes that grow or decline at a rate proportional to their current value. These functions are vital in many areas of calculus and statistics, particularly when modeling continuous growth or decay. The given problem's function \(\frac{e^{x}}{1+e^{2x}}\) arises from an exponential function, but is tamed by the denominator to model a bounded growth situation similar to logistic growth. To integrate such functions, you might need to manipulate the equation to reveal a more familiar pattern or apply integrating techniques suited for rational functions involving exponentials.
Area Under a Curve
The idea of finding the area under a curve is a fundamental application of integration. When we calculate the definite integral of a function, we are effectively measuring the space under the curve and above the x-axis over a specific interval. The area can represent distances, probabilities, quantities, or other physical concepts, depending on the function's context. In the exercise, translating the given function into a solvable form revealed that the integral corresponds to the natural logarithm function. This transformation made the complex exponential curve more approachable, allowing for the easy determination of the region's area.

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

Evaluate the integral. $$ \int_{0}^{1} \cosh ^{2} x d x $$

Chemical Reactions Chemicals A and B combine in a 3-to-1 ratio to form a compound. The amount of compound \(x\) being produced at any time \(t\) is proportional to the unchanged amounts of \(\mathrm{A}\) and \(\mathrm{B}\) remaining in the solution. So, if 3 kilograms of \(\mathrm{A}\) is mixed with 2 kilograms of \(\mathrm{B}\), you have \(\frac{d x}{d t}=k\left(3-\frac{3 x}{4}\right)\left(2-\frac{x}{4}\right)=\frac{3 k}{16}\left(x^{2}-12 x+32\right)\) One kilogram of the compound is formed after 10 minutes. Find the amount formed after 20 minutes by solving the equation \(\int \frac{3 k}{16} d t=\int \frac{d x}{x^{2}-12 x+32}\)

Find any relative extrema of the function. Use a graphing utility to confirm your result. $$ h(x)=2 \tanh x-x $$

Solve the differential equation. $$ \frac{d y}{d x}=\frac{1-2 x}{4 x-x^{2}} $$

Discuss several ways in which the hyperbolic functions are similar to the trigonometric functions.
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Probability and Statistics

Read Explanation

Mechanics Maths

Read Explanation

Decision Maths

Read Explanation

Geometry

Read Explanation

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.