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Chapter 5: Problem 8

The integrand of the definite integral is a difference of two functions. Sketch the graph of each function and shade the region whose area is represented by the integral. $$ \int_{-\pi / 4}^{\pi / 4}\left(\sec ^{2} x-\cos x\right) d x $$

Short Answer

Expert verified
The sketch should show two curves representative of \( \sec^{2}x \) and \( \cos{x} \) between -蟺/4 and 蟺/4. The shaded region between these two curves represents the integral given.

Step by step solution

01

Graph of \( \sec^{2}x \)

Firstly, sketch the graph of \( \sec^{2}x \) which is the same as \( 1 / \cos^{2}x \) on the interval [-蟺/4, 蟺/4]. Please note that \( \sec^{2}x \) is always greater than or equal to 1.
02

Graph of \( \cos{x} \)

After graphing the first function, let's sketch the second function \( \cos{x} \) on the same interval [-蟺/4, 蟺/4]. Keep in mind that for this interval, \( \cos{x} \) declines from \/2\/2 to \/2\/2.
03

Shading the Area

Now that both functions are graphed on the same interval, shade the area between them. This shaded region represents the integral of the difference of these two functions on the given interval.

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Most popular questions from this chapter

Irrigation Canal Gate The vertical cross section of an irrigation canal is modeled by \(f(x)=\frac{5 x^{2}}{x^{2}+4}\) where \(x\) is measured in feet and \(x=0\) corresponds to the center of the canal. Use the integration capabilities of a graphing utility to approximate the fluid force against a vertical gate used to stop the flow of water if the water is 3 feet deep.

The centroid of the plane region bounded by the graphs of \(y=f(x), y=0, x=0,\) and \(x=1\) is \(\left(\frac{5}{6}, \frac{5}{18}\right)\). Is it possible to find the centroid of each of the regions bounded by the graphs of the following sets of equations? If so, identify the centroid and explain your answer. (a) \(y=f(x)+2, y=2, x=0,\) and \(x=1\) (b) \(y=f(x-2), y=0, x=2,\) and \(x=3\) (c) \(y=-f(x), y=0, x=0,\) and \(x=1\) (d) \(y=f(x), y=0, x=-1,\) and \(x=1\)

Determine which value best approximates the volume of the solid generated by revolving the region bounded by the graphs of the equations about the \(x\) -axis. (Make your selection on the basis of a sketch of the solid and not by performing any calculations.) \(y=e^{-x^{2} / 2}, \quad y=0, \quad x=0, \quad x=2\) (a) 3 (b) -5 (c) 10 (d) 7 (e) 20

(a) use a graphing utility to graph the region bounded by the graphs of the equations, (b) explain why the area of the region is difficult to find by hand, and (c) use the integration capabilities of the graphing utility to approximate the area to four decimal places. $$ y=x^{2}, \quad y=\sqrt{3+x} $$

The area of the region bounded by the graphs of \(y=x^{3}\) and \(y=x\) cannot be found by the single integral \(\int_{-1}^{1}\left(x^{3}-x\right) d x\). Explain why this is so. Use symmetry to write a single integral that does represent the area.
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