Question

Graphical Reasoning 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 integral of the function \(x \sqrt{x^{2}+1}\) from -2 to 2 is zero.

Step-by-step solution

Understanding the integrand

The first step is to understand the integrand, which is \(x \sqrt{x^{2}+1}\). This integrand involves the product of \(x\) and \(\sqrt{x^{2}+1}\).

Graphing the integrand

Next is to graph the integrand. To do this, you will need a graphing utility. You may use any graphing utility you are comfortable with. The integrand \(x \sqrt{x^{2}+1}\) will produce a figure that is symmetric about the y-axis.

Determining the sign of the integral

To determine whether the definite integral is positive, negative, or zero, we need to examine the area under the curve from -2 to 2. Because of the aforementioned symmetry, the area under the curve from -2 to 0 will exactly cancel out the area from 0 to 2. Therefore, the integral is zero.