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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 6.3.9c
Chapter 6, Problem 6.3.9c

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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1
Step 1: Identify the region R and the axis of rotation. The region R is revolved about the line y=1, which is a horizontal line above the x-axis. This indicates that the method of washers or shells may be appropriate for calculating the volume.
Step 2: Determine the method to use. Since the axis of rotation is horizontal, the washer method is typically used when the region is bounded by curves and the volume is calculated by subtracting the inner radius from the outer radius.
Step 3: Express the radii in terms of the variable of integration. The outer radius is the distance from y=1 to the outer curve, and the inner radius is the distance from y=1 to the inner curve. These distances should be expressed as functions of x or y, depending on the orientation of the region.
Step 4: Write the volume integral using the washer method formula: \( V = \pi \int_{a}^{b} \left[ R_{\text{outer}}^2 - R_{\text{inner}}^2 \right] \, dx \), where \( R_{\text{outer}} \) and \( R_{\text{inner}} \) are the radii expressed as functions of x, and \( a \) and \( b \) are the bounds of integration.
Step 5: Substitute the specific expressions for \( R_{\text{outer}} \) and \( R_{\text{inner}} \) into the integral, ensuring that the bounds of integration correspond to the region R. Simplify the integral expression to finalize the setup for calculating the volume.

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Key Concepts

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

Solid of Revolution

A solid of revolution is a three-dimensional shape created by rotating a two-dimensional region around a straight line (axis of rotation). The volume of such solids can be calculated using integral calculus, specifically through methods like the disk method or the washer method, depending on the shape of the region and the axis of rotation.
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Finding Volume Using Disks

Volume Integral

The volume of a solid of revolution can be determined using volume integrals, which involve integrating the area of cross-sections perpendicular to the axis of rotation. For a region revolved around a horizontal line, the volume can be expressed as an integral of the form V = π ∫ [f(x) - k]^2 dx, where f(x) is the function defining the region and k is the line of rotation.
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Finding Volume Using Disks

Washer Method

The washer method is a technique used to find the volume of a solid of revolution when the region being revolved has a hole in the middle, resembling a washer. This method involves subtracting the volume of the inner solid from the volume of the outer solid, leading to an integral that accounts for both the outer and inner radii of the washers formed during the revolution.
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Euler's Method
Related Practice
Textbook Question

Oscillating growth rates Some species have growth rates that oscillate with an (approximately) constant period P. Consider the growth rate function N'(t) = r+A sin 2πt/P, where A and r are constants with units of individuals/yr, and t is measured in years. A species becomes extinct if its population ever reaches 0 after t=0.


c. Suppose P=10, A=50, and r=5. If the initial population is N(0)=10, does the population ever become extinct? Explain.

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Textbook Question

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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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

13–16. Displacement from velocity Consider an object moving along a line with the given velocity v. Assume time t is measured in seconds and velocities have units of m/s.


c. Find the distance traveled over the given interval.


v(t) = 4t³ - 24t²+20t on [0, 5]

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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

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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