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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 6.3.5c
Chapter 6, Problem 6.3.5c

Let R be the region bounded by the curve y=√cos x and the x-axis on [0, π/2]. A solid of revolution is obtained by revolving R about the x-axis (see figures). 


c. Write an integral for the volume of the solid.

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Identify the region R bounded by the curve \(y = \sqrt{\cos x}\) and the x-axis on the interval \([0, \frac{\pi}{2}]\). This means the region lies between \(y = 0\) and \(y = \sqrt{\cos x}\) for \(x\) in \([0, \frac{\pi}{2}]\).
Since the solid is formed by revolving the region R about the x-axis, use the disk method to find the volume. The volume of a solid of revolution generated by revolving a curve \(y = f(x)\) about the x-axis from \(x = a\) to \(x = b\) is given by the integral \(V = \pi \int_a^b [f(x)]^2 \, dx\).
In this problem, the function is \(f(x) = \sqrt{\cos x}\), so the radius of each disk is \(\sqrt{\cos x}\). Squaring this radius gives the area of the cross-sectional disk as \([\sqrt{\cos x}]^2 = \cos x\).
Set up the integral for the volume using the limits of integration \(0\) to \(\frac{\pi}{2}\):
\[V = \pi \int_0^{\frac{\pi}{2}} \cos x \, dx.\]

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Key Concepts

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

Region Bounded by a Curve and the x-axis

Understanding the region bounded by the curve y = √cos x and the x-axis on [0, π/2] is essential. This involves identifying the area under the curve from x = 0 to x = π/2, which forms the cross-sectional shape to be revolved around the x-axis.
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Finding Area Between Curves on a Given Interval

Volume of a Solid of Revolution Using the Disk Method

The disk method calculates the volume of a solid formed by revolving a region around the x-axis. The volume is found by integrating π times the square of the radius function (here, y = √cos x) with respect to x over the given interval.
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Setting up Definite Integrals for Volume

Writing the integral requires expressing the volume as an integral with proper limits and integrand. For this problem, the integral is from 0 to π/2 of π times (√cos x)² dx, simplifying to π∫₀^{π/2} cos x dx, which represents the volume of the solid.
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Definition of the Definite Integral
Related Practice
Textbook Question

Where do they meet? Kelly started at noon (t=0) riding a bike from Niwot to Berthoud, a distance of 20 km, with velocity v(t) = 15 / (t + 1)² (decreasing because of fatigue). Sandy started at noon (t=0) riding a bike in the opposite direction from Berthoud to Niwot with velocity u(t) = 20 / (t + 1)² (also decreasing because of fatigue). Assume distance is measured in kilometers and time is measured in hours.


c. When do they meet? How far has each person traveled when they meet?

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Textbook Question

Region R is revolved about the line y=1 to form a solid of revolution.


c. Write an integral for the volume of the solid.

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Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

c. The work required to lift a 10-kg object vertically 10 m is the same as the work required to lift a 20-kg object vertically 5 m.

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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.


c. What is the cardiac output over a period of 1 min? (Use calculus; then check your answer with algebra.)

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Textbook Question

Depletion of natural resources Suppose r(t) = r0e^−kt, with k>0, is the rate at which a nation extracts oil, where r0=10⁷ barrels/yr is the current rate of extraction. Suppose also that the estimate of the total oil reserve is 2×10⁹ barrels. 


c. Find the minimum decay constant k for which the total oil reserves will last forever.

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Textbook Question

9–10. Velocity graphs The figures show velocity functions for motion along a line. Assume the motion begins with an initial position of s(0)=0. Determine the following.

c. The position at t=5

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