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. $$ \int_{-2}^{2} x \sqrt{2-x} d x $$

Short answer

The definite integral of \(x\sqrt{2 - x}\) from -2 to 2 is zero since the function is symmetrical about the y-axis and the negative and positive areas underneath the curve cancel each other out.

Step-by-step solution

Understand the Integrand Function

The function given is \(x \sqrt{2 - x}\). This function is a product of \(x\) and \(\sqrt{2 - x}\). It should be noted that the square root is only defined when \(2 - x \geq 0\) or \(x \leq 2\). This limits the x-values to \(x \leq 2\).

Graph the Integrand

We can graph the function \(x\sqrt{2-x}\) using a graphing utility. This would give us an idea of the shape of the function and especially the areas below and above the x-axis between -2 and 2.

Determine the Sign of the Definite Integral

Notice that the resulting graph is symmetrical about the y-axis. This implies that the function will have the same areas under the curve but opposite signs on the intervals \(-2 \leq x < 0\) and \(0 < x \leq 2\). Since the integrand is negative in the interval \(-2 \leq x \leq 0\) and positive in the interval \(0 \leq x \leq 2\) because of the square root sign over \(2 - x\), the areas cancel each other out when evaluating the integral from -2 to 2.