Question

Find the area of the region bounded by the graphs of the equations. Use a graphing utility to graph the region and verify your result. $$ y=x e^{-x^{2} / 4}, y=0, x=0, x=\sqrt{6} $$

Short answer

The area of the region bounded by the given equations is approximately 1.213 sq.units.

Step-by-step solution

Understand the boundaries of the region

The given problem describes a region bounded by the equations \(y=x e^{-x^{2} / 4}\), \(y=0\), \(x=0\), and \(x=\sqrt{6}\). The functions \(y=0\), \(x=0\), and \(x=\sqrt{6}\) form a rectangle from the origin to x=\(\sqrt{6}\), on the x-axis. This will be the region where the integration will happen.

Set up the integral to calculate the area

To calculate the area under the curve, an integral of the function between the defined limits must be calculated. In this case, the integral becomes \(\int_{0}^{\sqrt{6}} x e^{-x^2/4} dx\).

Evaluate the integral

Now we need to solve the integral. This can be done using substitution method. Here, let \(u = -x^2/4\). So, \(du =- x/2 dx\). Then the integral will reduce to -2 times \(\int e^u du\) from \(0\) to \(-3/2\). The integral of \(e^u\) is \(e^u\). Thus, evaluating the definite integral, we get \(-2 [e^{u}]_{0}^{-3/2}\), which equals to \(-2(e^{-3/2}-1)\).

Evaluation and simplification of the area

The answer obtained in the Step 3 would be in terms of 'e'. Further simplifying it for a more understandable answer, \(-2(e^{-3/2}-1)\) equals approximately 1.213.

Graphing the function

A graphing calculator or software can be used to draw the function \(y = x e^{-x^{2} / 4}\) and to confirm the obtained result visually.