Question

Use a graphing utility to graph the integrand. Use the graph to determine whether the definite integral is positive, negative, or zero. $$ \int_{-2}^{2} x \sqrt{x^{2}+1} d x $$

Short answer

The definite integral \( \int_{-2}^{2} x \sqrt{x^{2}+1} d x \) is zero.

Step-by-step solution

Determine the function to be Integrated

The function that is being integrated is \( f(x) = x \sqrt{x^{2}+1} \). This is the function that will be graphed.

Graph the Function

Plot the function \( f(x) = x \sqrt{x^{2}+1} \) in a graphing utility. Pay special attention to the range of the x-values. Here, we are interested in the interval from -2 to 2.

Examine the graph

Examine the graph over the interval from -2 to 2. If the graph lies above the x-axis in this interval, then the definite integral will be positive. If it lies below the x-axis, then the definite integral will be negative. If it's exactly on the x-axis, the integral will be zero. In this case, one can observe that the graph lies both above and below the x-axis, with the area below the x-axis matching the area above, indicating that the overall integral will be zero.