Question

(a) use a graphing utility to graph the region bounded by the graphs of the equations, \((b)\) find the area of the region, and (c) use the integration capabilities of the graphing utility to verify your results. $$ f(x)=1 /\left(1+x^{2}\right), \quad g(x)=\frac{1}{2} x^{2} $$

Short answer

To find the area between the curves \(f(x) = \frac{1}{1+x^2}\) and \(g(x) = \frac{1}{2} x^2\), the integral \(\int_{-1}^{1}[\frac{1}{1+x^2}-\frac{1}{2} x^2] dx\) must be evaluated. The result can be confirmed by integrating the functions within the same bounds of -1 and 1, using a graphing utility's integration feature.

Step-by-step solution

Plotting the graphs

Using a graphing utility, plot the functions \(f(x) = \frac{1}{1 + x^2}\) and \(g(x) = \frac{1}{2} x^2\). Observe where these functions intersect each other, as this will be important for the integration step.

Find the area between the curves

The area between the two curves is given by the definite integral \(A = \int [(f(x) - g(x)] dx\) over the interval where the curves intersect. The points of intersection need to be identified by setting the two equations equal to each other and solving.

Set the equations equal to each other

\(f(x) = g(x)\) gives \(\frac{1}{1 + x^2} = \frac{1}{2} x^2\). By cross multiplying and simplifying, one arrives at \(2 = x^4+x^2\) which is rearranged as \(x^4 + x^2 - 2 = 0\). This quadratic in \(x^2\) can be factored into \((x^2-1)(x^2+2)=0\), yielding \(x = 1, -1\) as solutions.

Evaluate the definite integral

Evaluate \(A = \int_{-1}^{1} [(f(x) - g(x)] dx = \int_{-1}^{1}[\frac{1}{1+x^2}-\frac{1}{2} x^2] dx\). Solving this definite integral will provide the area of the region between the curves.

Verify the results using a graphing utility

Use the integral feature of the graphing utility to evaluate the definite integral figured out in theprevious step. If done correctly, the calculated area should match up with the area obtained from graphic tool.