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Ch. 6 - Applications of Integration
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 6 - Applications of IntegrationProblem 6.2.65b
Chapter 6, Problem 6.2.65b

Determine whether the following statements are true and give an explanation or counterexample.


b. The area of the region between y=sin x and y=cos x on the interval [0,π/2] is ∫π/20(cosx−sinx)dx.

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First, identify the two functions given: \(y = \sin x\) and \(y = \cos x\) on the interval \([0, \frac{\pi}{2}]\).
Determine which function is on top (greater) and which is on the bottom over the interval \([0, \frac{\pi}{2}]\). This is important because the area between two curves is found by integrating the difference between the top function and the bottom function.
Evaluate the values of \(\sin x\) and \(\cos x\) at the endpoints: at \(x=0\), \(\sin 0 = 0\) and \(\cos 0 = 1\), so \(\cos x\) is greater; at \(x=\frac{\pi}{2}\), \(\sin \frac{\pi}{2} = 1\) and \(\cos \frac{\pi}{2} = 0\), so \(\sin x\) is greater. This means the two curves cross somewhere in the interval.
Find the point where \(\sin x = \cos x\) in \([0, \frac{\pi}{2}]\) by solving \(\sin x = \cos x\). This will give the exact point where the top and bottom functions switch.
To find the total area between the curves, split the integral at the crossing point and integrate the absolute difference accordingly: \(\int_0^{c} (\cos x - \sin x) \, dx + \int_c^{\frac{\pi}{2}} (\sin x - \cos x) \, dx\), where \(c\) is the crossing point. Therefore, the statement that the area is \(\int_0^{\frac{\pi}{2}} (\cos x - \sin x) \, dx\) without splitting is not correct.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Definite Integral as Area Between Curves

The definite integral of the difference between two functions over an interval gives the net area between their graphs. If one function is always above the other, the integral of (top function - bottom function) over that interval equals the area between them.
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Finding Area Between Curves on a Given Interval

Determining Which Function is on Top

To correctly set up the integral for area, identify which function is greater on the interval. For y = sin x and y = cos x on [0, π/2], compare values to see which curve lies above, ensuring the integrand is the top function minus the bottom function.
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Determining Concavity Given a Function

Evaluating the Integral and Sign of the Result

The integral ∫ (cos x - sin x) dx over [0, π/2] may yield a positive or negative value depending on which function is on top. A negative result indicates the order of subtraction is reversed, so the absolute value or swapping the order is needed to represent the area.
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Integrals Resulting in Natural Logs
Related Practice
Textbook Question

Distance traveled and displacement Suppose an object moves along a line with velocity (in ft/s) v(t)=6−2t, for 0≤t≤6, where t is measured in seconds.


b. Find the displacement of the object on the interval 0≤t≤6.

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Textbook Question

Two runners At noon (t=0), Alicia starts running along a long straight road at 4 mi/hr. Her velocity decreases according to the function v(t) = 4 / t + 1 for t≥0. At noon, Boris also starts running along the same road with a 2-mi head start on Alicia; his velocity is given by u(t) = 2 / t + 1, for t≥0. Assume t is measured in hours.


b. When, if ever, does Alicia overtake Boris?

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Textbook Question

Filling a tank A 2000-liter cistern is empty when water begins flowing into it (at t=0 at a rate (in L/min) given by Q′(t) = 3√t, where t is measured in minutes.


b. Find the function that gives the amount of water in the tank at any time t≥0.

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Textbook Question

A torus (doughnut) A torus is formed when a circle of radius 2 centered at (3, 0) is revolved about the y-axis.


b. Use the washer method to write an integral for the volume of the torus.

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Textbook Question

Blood flow A typical human heart pumps 70 mL of blood (the stroke volume) with each beat. Assuming a heart rate of 60 beats/min (1 beat/s), a reasonable model for the outflow rate of the heart is V′(t)=70(1+sin 2πt), where V(t) is the amount of blood (in milliliters) pumped over the interval [0,t],V(0)=0 and t is measured in seconds.


b. Find the function that gives the total blood pumped between t=0 and a future time t>0.

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Textbook Question

55–58. Marginal cost Consider the following marginal cost functions.


b. Find the additional cost incurred in dollars when production is increased from 500 units to 550 units.


C′(x) = 300+10x−0.01x²

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