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Chapter 5: Problem 26

Sketch and find the area of the region bounded by the given curves. Choose the variable of integration so that the area is written as a single integral. $$y=\frac{\ln x}{x}, y=\frac{1-x}{x^{2}+1}, x=4$$

Short Answer

Expert verified
To find the exact area enclosed, one needs to solve the integral. This would require calculus. Just to remind that the integral might not be solvable analytically, and one might need to use numerical approximation methods.

Step by step solution

01

Sketch the Graphs

First, plot the graph of the functions \(y=\frac{\ln x}{x}\), \(y=\frac{1-x}{x^{2}+1}\), and \(x=4\). The graphs should cross at some points, defining the area enclosed.
02

Determine the limits of Integration

The limits of integration are determined by the intersection points of the graphs of the functions. You can find the intersection points by setting the two curves equal to each other and solve for \(x\). For this task, since \(x=4\) is given, it should be one of the boundaries. The other boundary point is obtained by solving \(\frac{\ln x}{x} = \frac{1-x}{x^{2}+1}\).
03

Set up the Integral

After identifying the limits of integration, the next step is to set up the integral. This involves subtracting one function from the other, from left to right on the x-axis. We integrate the absolute difference of the two function within the obtained limits to find the area. This gives \(\int_a^b |f(x) - g(x)| dx\), where \(a\) and \(b\) are the obtained limits of integration.
04

Evaluate the Integral

The last step involves evaluating the definite integral. Use either the fundamental theorem of calculus or numerical integration methods to find the area under the curve within the given limits.

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Key Concepts

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

Integration
Integration is a fundamental concept in calculus, used to find the accumulation of quantities. It can be understood as the reverse operation of differentiation. The process of integration is primarily concerned with finding the area under a curve. Imagine plotting a curve on a graph, the shaded area under this curve represents the integral.

This concept is essential in determining quantities where the rate of change is known. In our exercise, we need to integrate to find the area between two curves. Integration takes these curves and allows us to calculate the total accumulated area under them.
  • The integral sign (鈭�) is used to denote the integration process.
  • The process translates the curve into a number, representing the area under or between curves.
Definite integral
A definite integral can be thought of as a tool to calculate the exact area under a curve between two points. Mathematically, it is expressed as \(\int_a^b f(x) \, dx \), where \(a\) and \(b\) are the limits of integration. This confines our area of interest to a specific segment of the x-axis.

In the given exercise, we need a definite integral to find the area bounded by the curves. We identified the limits of integration by determining where the curves intersect.
  • The lower limit \(a\) and the upper limit \(b\) designate the boundary points of the integral.
  • A definite integral gives a fixed numerical value, representing the area between the curve and the x-axis from \(a\) to \(b\).
Area of a region
Calculating the area of a region bounded by curves involves integrating the absolute difference of two functions. In our exercise, once the curves intersect, they form a closed region. We need to determine which function is above the other over the interval.

To find the area, integrate the difference between the top function and the bottom function across the interval defined by the intersection points. This is expressed as \(\int_a^b |f(x) - g(x)| \, dx \). As curves may switch positions, always ensure the upper curve is subtracted by the lower one.
  • The area is always a non-negative value, as denoted by the absolute difference.
  • Switching which function is subtracted might be required over different intervals.
Intersection points
Intersection points are where two curves meet or cross each other, and are essential in determining the limits for a definite integral. In the exercise, setting the two functions equal helps us find these locations: \(\frac{\ln x}{x} = \frac{1-x}{x^2+1}\).

Solve this equation to find the value(s) of \(x\) at these intersections. Including \(x=4\), given directly in the problem, makes it one of the boundaries.
  • Intersection points define the intervals over which integration is performed.
  • Graphically, these are the points where curves cross, marking the start or end of a bounded region.
  • Accurate solutions for intersection points ensure the correct area calculation.

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