Question

Graphical Reasoning In Exercises \(1-4,\) use a graphing utility to graph the integrand. Use the graph to determine whether the definite integral is positive, negative, or zero. $$ \int_{0}^{\pi} \frac{4}{x^{2}+1} d x $$

Short answer

The definite integral of the function \(y = \frac{4}{x^{2}+1}\) from \(0\) to \(\pi\) is positive.

Step-by-step solution

- Plot the function

Plot the function \(y= \frac{4}{x^{2}+1}\) across the domain \(0\) to \(\pi\) using a graphing utility. Initially, the graph seems to be always above the x-axis i.e., always positive.

- Check for negative areas

Look for the portions of the graph that lie below the x-axis between \(0\) and \(\pi\). The definite integral is negative if the majority of the area is below the x-axis. However, as it can be seen the graph does not drop below the x-axis.

- Determine the Sign of the Integral

Since the entire part of the graph lies above x-axis. We can conclude that the entire integral is positive.