Problem 46 - Essential Calculus

Question

In Exercises \(45-48\), use a graphing utility to graph the region bounded by the graphs of the equations. Use the integration capabilities of the graphing utility to approximate the centroid of the region. $$ y=x e^{-x / 2}, y=0, x=0, x=4 $$

Step-by-Step Solution

Answer

We can use the feature of the graphing utility to find the centroid of the area under the curve \(y = xe^{-x / 2}\) within the limits x=0 and x=4, the result of which will be an ordered pair representing the coordinates of the centroid.

Step 1: Plot the Graphs

Plot the three functions \(y = xe^{-x / 2}\), \( y = 0\), and \( x = 0\) on the graphing utility. Remember, \(x = 4\) is the upper limit.

Step 2: Identify the Bounded Region

The region is defined where all graphs intersect and form a closed unbiased area. Here, this area is defined by the x-axis (y=0), the y-axis (x=0), and the curve \(y = xe^{-x / 2}\) between x=0 and x=4.

Step 3: Calculate the Centroid

Centroid (x̄ , ȳ) of an area can be found using the formulae \(x̄ = \frac{1}{A}\int_{0}^{4} x y dx\) and \(ȳ = \frac{1}{2A}\int_{0}^{4} y^2 dx\) where \(A = \int_{0}^{4} y dx\) is the total area under the curve. Using the graphing utility's integration capabilities, perform this operation to approximate the centroid of the region.

Key Concepts

Region Bounded by Curves

Calculating the area bounded by curves involves understanding the overlap or enclosed space formed by these curves. In this exercise, the focus is on identifying the region bounded by the curve \(y = xe^{-x / 2}\), the x-axis (\(y = 0\)), the y-axis (\(x = 0\)), and the line \(x = 4\). To find such a region, you first graph these functions.

• \(y = 0\) represents the x-axis.
• \(x = 0\) represents the y-axis.
• \(x = 4\) is a vertical line cutting the plane perpendicular to the x-axis.
• \(y = xe^{-x / 2}\) is an exponentially decaying function.

By plotting these on a graphing utility, you can see where they intersect and enclose an area. The bounded region is the shape enclosed by these boundaries, visually identifiable in a properly scaled graph.

Centroid Calculation

The centroid is the geometric center or the 'balancing point' of a shape. For a planar region, it can be thought of as a weighted balance of the entire area, located at coordinates \((\bar{x}, \bar{y})\). Calculating the centroid requires integration to find these coordinates.
For the x-coordinate of the centroid (\(\bar{x}\)), use the formula: \[\bar{x} = \frac{1}{A}\int_{0}^{4} x \cdot y \, dx\] For the y-coordinate (\(\bar{y}\)), use: \[\bar{y} = \frac{1}{2A}\int_{0}^{4} y^2 \, dx\]Here, \(A\) is the total area under the curve found using:\[A = \int_{0}^{4} y \, dx\]By performing these integrations using graphing utilities, the precise centroid is approximated, offering insights into the balance of the region.

Graphing Utility Integration

A graphing utility enables efficient calculation of complex mathematical operations like integration, essential for solving calculus problems accurately. It is especially beneficial for approximating areas and centroids in bounded regions.
To use a graphing utility:

• Input the function \(y = xe^{-x / 2}\), along with the limits \(x = 0\) to \(x = 4\).
• Use the integration feature to calculate the area \(A\), and subsequently, the integrals for \(\bar{x}\) and \(\bar{y}\).
• Graph the functions visually to ensure the correct region is considered.

This allows you to seamlessly combine visual learning with computation, ensuring accurate and efficient problem-solving.

Exponential Functions

Exponential functions like \(y = xe^{-x / 2}\) have distinctive characteristics, such as rapid growth or decay. In the context of calculus, they are often used to model rates of change and decay processes.
Key features of exponential functions include:

• An initial value or multiplicative factor that affects the growth or decay rate.
• A base rate (\(e\) or Euler's number), making the function continuous and smooth.
• Transformations that shift or stretch the graph in various directions.

Understanding these functions, especially when combined with polynomials (as in this exercise), is crucial for analyzing dynamic systems and calculating regions and centroids.