Problem 4 - Essential Calculus

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

Step-by-Step Solution

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 1: 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 \(x \leq 2\).

Step 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.

Step 3: 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.

Key Concepts

Graphical Reasoning

Graphical reasoning involves looking at a visual representation of a function to draw conclusions about its behavior over a certain interval. In the context of definite integrals, graphs can help us understand the area under the curve, which represents the integral's value on that interval. For the function \(x \sqrt{2-x}\), using graphical reasoning involves plotting the function and observing its shape, behavior, and where it lies relative to the x-axis. By visualizing the integrand and identifying key features like symmetry, you can infer properties about the integral without starting with calculus. For example, seeing a symmetrical curve can hint at equal areas on either side of the y-axis.

Integrand Function

The integrand function in a definite integral is the function inside the integral sign that you'll be analyzing or graphing. Here, it's \(x \sqrt{2-x}\). To understand this function, break it down into its components:

• The term \(x\) serves as a linear multiplier affecting the output shape and direction.
• \(\sqrt{2-x}\) is a square root function, defined only for \(2 - x \geq 0\), meaning it only exists from \(-\infty\) to \(2\). This implies the function is only meaningful up to \(x = 2\).

Together, these factors determine the function's behavior between the specified limits, and it is crucial to comprehend the integrand before exploring further with a graph.

Graphing Utility

A graphing utility is a tool that lets students and educators visualize mathematical functions, making it easier to explore properties like symmetry, intercepts, and integrals. Using graphing software or a graphing calculator, you can plot \(x \sqrt{2-x}\), observing how it behaves between the limits \([-2, 2]\). Graphing tools are especially valuable for calculus beginners because they provide an intuitive visual representation:

• They display curves, showing where the function is positive or negative.
• They can highlight symmetry, suggesting how the areas under the curve might relate to each other.

By configuring a graphing utility to plot the integrand, you get a clearer understanding of the integral's properties before tackling complex calculations.

Symmetry in Calculus

Symmetry is a fascinating concept in calculus, often simplifying problems related to integrals. When dealing with the function \(x \sqrt{2-x}\), observing symmetry helps determine the behavior of the integral over symmetric intervals, like from \(-2\) to \(2\). If the graph is symmetric about a particular axis:

• Areas on either side that mirror each other can imply cancelation in definite integrals.
• This is because the positive area on one side is often balanced by a negative area of equal magnitude on the opposite side.

Identifying symmetry allows for an intuitive understanding of why and how certain integrals, such as those with equal areas above and below the x-axis, can evaluate to zero. Students can leverage symmetry to streamline calculation and grasp integral concepts more effectively.