Problem 46 - Calculus of a Single Variable

Question

Finding the Area of a Region In Exercises \(43-46,(\text { a) use }\) a graphing utility to graph the region bounded by the graphs of the equations, (b) find the area of the region, and (c) use the integration capabilities of the graphing utility to verify your results. $$ g(x)=\frac{4 \ln x}{x}, \quad y=0, \quad x=5 $$

Step-by-Step Solution

Answer

The area of the region under the curve of the function \( g(x)=\frac{4 \ln x}{x} \) between the x-axis and \( x=5 \) is \( 4 \ln 5 - 4 \) square units.

Step 1: Graph the Function

First, we should graph the function \( g(x)=\frac{4 \ln x}{x} \) alongside \( y=0 \) and at \( x=5 \). This will help to visualize the covered region. For the function \( g(x)=\frac{4 \ln x}{x} \), the domain is \( x > 0 \). According to this domain and the given value \( x=5 \), we are interested in the region starting from \( x=1 \) up to \( x=5 \).

Step 2: Calculate the Integral

The area under the curve of a function is found by integrating the function over the interval of interest. In this exercise, we want to find the area of the region bounded by \( g(x) \), \( y=0 \) and \( x=5 \), so we need to integrate \( g(x) \) from 1 to 5: \[A = \int_1^5 g(x) dx = \int_1^5 \frac{4 \ln x}{x} dx \]

Step 3: Solve the Integral

The indefinite integral of \( f(x) = \frac{4 \ln x}{x} \), which is \( F(x) = 4(\ln x - 1) \), can be found using standard calculus methods. So, the definite integral from 1 to 5 is evaluated as: \[A = [4(\ln 5 - 1) - 4(\ln 1 - 1)] = 4 \ln 5 - 4\]

Step 4: Verify the Results

To verify the result, it can be compared with the one obtained through the integration capability of a graphic calculator or other computation software. The above integral should be evaluated using the definite integral calculator to compare results.

Key Concepts

Graphing Utility

A graphing utility is a powerful tool in calculus. It helps us visualize mathematical functions by plotting their graphs. When dealing with multiple functions or constraints, like in our task with multiple lines bounding a region, a graphing utility simplifies understanding.
Using a graphing utility, you can easily generate graphs of equations such as \(g(x)=\frac{4 \ln x}{x}\). This visual aspect is critical because it allows us to view the area we are estimating or calculating.
It can also plot the line \(y=0\) and the vertical boundary at \(x=5\). These visuals guide us in defining the exact region we want to analyze, which is crucial before proceeding with integration or other calculus methods.

Area Under Curve

The concept of finding the area under a curve is fundamental in calculus. This area helps determine how much space a function takes up between defined boundaries along the x-axis and usually another function or a line like \(y=0\).
In the exercise, the area under \(g(x)\) from \(x=1\) to \(x=5\) represents the space occupied by our function in that interval. This area under the curve equals the integral of the function over the specified range.
Understanding this concept helps in various practical applications such as physics to calculate displacement, in economics for predicting total cost, and more. It's an essential tool in analyzing mathematical models.
Visualizing this using a graphing utility first gives us a qualitative look at where the function lies and how it behaves between the limits of integration.

Definite Integrals

Definite integrals are a part of integral calculus that provide the exact area under a curve between two points. They not only help in quantifying the area but also solidify understanding of how a function behaves over an interval.
In our exercise, we set up a definite integral for \(g(x)=\frac{4 \ln x}{x}\) from \(x=1\) to \(x=5\). This is represented by the expression: \[A = \int_1^5 \frac{4 \ln x}{x} \, dx \]
Solving this integral gives the total area. It accounts for any accumulative growth or decay of the function within the designated interval.
The relevance of definite integrals is vast—they are used from simple area calculations to more complex integral-related principles in advanced math and science fields.

Calculus Methods

Various calculus methods come into play when solving integrals or interpreting their results. These methods often rely on understanding the basic principles of calculus, such as the rules of differentiation and integration.
For the indefinite integral of \(g(x)=\frac{4 \ln x}{x}\), we apply standard techniques to derive \(F(x) = 4(\ln x - 1)\). This integral transforms the complex relationship into something more manageable that serves as a building block for finding the definite integral.
Applying the fundamental theorem of calculus, we find the definite integral across our specified bounds, \([1, 5]\), yielding the area: \[A = [4(\ln 5 - 1) - 4(\ln 1 - 1)] = 4 \ln 5 - 4\]
These methods are not just limited to textbook problems. They underpin analytical approaches in advanced research and problem-solving contexts, creating a bridge between theoretical mathematics and real-world applications.