Problem 38 - Essential Calculus

Question

Find the area of the region bounded by the graphs of the equations. $$ y=\frac{x}{1+e^{x^{2}}}, y=0, x=2 $$

Step-by-Step Solution

Answer

The area is approximated by the integral \[A=\int_{{0}}^{{2}}\frac{x}{1+e^{x^{2}}}~dx\] It requires numerical methods to be solved.

Step 1: Understanding the problem

The problem provides the equations \(y=\frac{x}{1+e^{x^{2}}}\), \(y=0\), and \(x=2\). The graph of the function \(y=\frac{x}{1+e^{x^{2}}}\) lies above the x-axis between \(x=0\) and \(x=2\), while \(y=0\) represents the x-axis. So we are asked to find the area of the region between this curve and the x-axis, up to \(x=2\).

Step 2: Setting up the integral

To find the area bounded by the curve and the x-axis from \(x=0\) to \(x=2\), we will use the area formula: \[A=\int_{{a}}^{{b}}|f(x)|~dx\], where \(a\) and \(b\) are the points where the graph cuts the x-axis. In this case, \(a=0\), \(b=2\), and \(f(x)=\frac{x}{1+e^{x^{2}}}\)

Step 3: Calculating the area

Substituting the limits and the function into the area formula, we get: \[A=\int_{{0}}^{{2}}\left|\frac{x}{1+e^{x^{2}}}\right|~dx\]Since the function is positive in the range \(x=0\) to \(x=2\), we can remove the absolute signs. Now, the function to integrate has become: \[A=\int_{{0}}^{{2}}\frac{x}{1+e^{x^{2}}}~dx\] Since this integral is non elementary, we can use a numerical method like the Simpson's rule or a power series to approximate the area.