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. $$ y=x e^{-x / 2}, y=0, x=0, x=4 $$

Short 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-by-step solution

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.

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.

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.