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. $$ \int_{0}^{\pi} \cos x d x $$

Short answer

The definite integral of the function \(\cos x\) from 0 to \(\pi\) is zero.

Step-by-step solution

Plot the Function

Use a graphing utility to sketch the graph of the function \(y = \cos x\) on the interval [0, \(\pi\)]. Observe that the function starts from a maximum at \(x=0\) (since \(\cos0 = 1\)), descends to 0 at \(x = \(\pi/2\) and continues to decline until it reaches a minimum at \(x = \(\pi\) ) (since \(\cos \(\pi\) = -1)).

Observe the Graph

In the given range (0 to \(\pi\)), the graph of the cos x function lies both above and below the x-axis equally.

Determine the Sign of the Definite Integral

Given that the area under the curve is equal above and below the x-axis, from the graphical observation, we can infer the integral value over the given range is Zero.

Confirm with Calculation

It can be confirmed by manually integrating the function. The integral of \(\cos x\) is \(\sin x\), so evaluating \(\sin x\) from 0 to \(\pi\) results in \(\sin \(\pi\) - sin0 = 0 - 0 = 0\). This should correspond with what observed graphically.

Key concepts

Graphical Reasoning

Understanding a definite integral through graphical reasoning involves analyzing the graph of the integrand to make initial estimates or judgments about the integral's value. For example, when given the task of evaluating \[\int_{0}^{\pi} \cos x \, dx\], one of the first steps is to sketch or visualize the cosine function over the interval from 0 to \(\pi\).

Here's how graphical reasoning comes into play: the graph of \(y = \cos x\) starts at a maximum value of 1 when \(x = 0\) and ends at a minimum of -1 when \(x=\pi\), crossing the x-axis at \(x = \frac{\pi}{2}\). By observing this behavior, one can infer that the area under the curve from \(x = 0\) to \(x = \frac{\pi}{2}\) will be positive, and from \(x = \frac{\pi}{2}\) to \(x = \pi\) will be negative, suggesting these areas will cancel each other out. This cancellation implies that the integral will likely equal zero, though this needs to be confirmed through calculation. Therefore, through graphical reasoning, students can gain intuitive knowledge of what to expect from their calculations before performing them.

Graphing Utility

A graphing utility is a powerful tool that can be used to visualize mathematical functions and is particularly useful in the study of integrals. To graph the function \(y = \cos x\) over the interval [0, \(\pi\)], a student can use a variety of graphing tools, ranging from graphing calculators to computer software programs that provide a visual representation of the function.

A graphing utility can help in several ways:

• It provides a visual representation for better understanding the behavior of the cosine function.
• It assists in predicting whether the area under the curve will result in a positive, negative, or zero integral by showing where the curve lies in relation to the x-axis.
• It can also help in confirming results by comparing the graphical area with the calculated integral.

By using a graphing utility, students can avoid common missteps when evaluating integrals and greatly streamline the learning process.

Integral of Cosine

Integrating the cosine function is a fundamental concept in calculus. The integral of cosine is straightforward: it is the sine function. When calculating the definite integral \[\int_{0}^{\pi} \cos x \, dx\], we apply the fundamental theorem of calculus, which involves finding the antiderivative of the cosine function and evaluating it at the boundaries.

Here's the calculation process:

1. The antiderivative of \(\cos x\) is \(\sin x\).
2. To find the definite integral, we evaluate \(\sin x\) at the upper limit \(x = \pi\) and the lower limit \(x = 0\), and find the difference.
3. Since \(\sin(\pi) = 0\) and \(\sin(0) = 0\), the difference is 0, confirming that the integral evaluates to zero.

Through the combination of graphical reasoning and analytical calculation, the integral of cosine over the given interval can be confidently determined. Understanding how to integrate basic trigonometric functions like cosine is essential for students as they progress through more advanced topics in calculus.