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. Witch of Agnesi \(y=8 /\left(x^{2}+4\right), y=0, x=-2, x=2\)

Short answer

After following these steps, the coordinates of the centroid of the defined region can be found. As it depends on the graphing calculator, a specific answer cannot be provided.

Step-by-step solution

Graph the Region

Plug in the given equations \(y=8 /\left(x^{2}+4\right)\), \(y=0\), \(x=-2\), and \(x=2\) into a graphing utility. Identify the region bounded by these equations.

Calculate Area

Use the integration capabilities of the utility to evaluate the definite integral \(A=\int_{-2}^{2} 8 / (x^{2} + 4)\,dx\). This represents the area of the region.

Find Centroid

The x-coordinate of the centroid is given by \(x_{c}= \frac{1}{A}\int_{-2}^{2} x\cdot[8 / (x^{2} + 4)]\,dx\) and the y-coordinate is given by \(y_{c}= \frac{1}{A}\int_{-2}^{2} [8 / (x^{2} + 4)]^{2}\,dx\). Calculate these integral values using the graphing calculator.