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Chapter 4: Problem 95

Use a graphing utility to evaluate the integral. Graph the region whose area is given by the definite integral. $$ \int_{0}^{3}\left(\theta+\cos \frac{\theta}{6}\right) d \theta $$

Short Answer

Expert verified
To solve this, use a graphing utility and input the function \(\theta + \cos\left(\frac{\theta}{6}\right)\), and ask it to compute the definite integral from \(\theta = 0\) to \(\theta = 3\). The number it gives represents the area under the curve of the function within the set limits, which is visually represented on the graph by the shaded area underneath the curve between these points.

Step by step solution

01

Insert Function into Graphing Utility

Use a graphing utility, such as Desmos, and insert the function \(\theta + \cos\left(\frac{\theta}{6}\right)\). Make sure your graphing tool is set to integrate over the correct range, which is \(\theta = 0\) to \(\theta = 3\).
02

Evaluate the Integral

Ask the utility to compute the definite integral over the specified range. This will give the numerical value for the area under the curve.
03

Graph the Region

Graph the function \(\theta + \cos\left(\frac{\theta}{6}\right)\) within the interval to visually see the region under the graph whose area is represented by the definite integral. Remember to shade the area beneath the curve, between the curve and x-axis, from \(\theta = 0\) and \(\theta = 3\).
04

Interpret the Result

The numerical outcome represents the area under the curve of the function \(\theta + \cos\left(\frac{\theta}{6}\right)\) within the limits \(\theta = 0\) and \(\theta = 3\). The area on the graph is the graphical representation of this information.

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Most popular questions from this chapter

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