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Ch. 6 - Applications of Integration
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 6 - Applications of IntegrationProblema 6.3.7b
Capitolo 6, Problema 6.3.7b

Use the region R that is bounded by the graphs of y=1+√x,x=4, and y=1 complete the exercises.


Region R is revolved about the x-axis to form a solid of revolution whose cross sections are washers.


b. What is the inner radius of a cross section of the solid at a point x in [0, 4]?

Guida verificata passo dopo passo
1
Step 1: Understand the problem. The region R is bounded by the graphs of y = 1 + √x, x = 4, and y = 1. When this region is revolved about the x-axis, the resulting solid has cross sections that are washers. The goal is to determine the inner radius of a washer at a point x in the interval [0, 4].
Step 2: Recall the formula for the inner radius of a washer. The inner radius is the distance from the x-axis to the inner curve of the region being revolved. In this case, the inner curve is the horizontal line y = 1.
Step 3: Analyze the geometry of the region. At any point x in [0, 4], the inner radius is determined by the vertical distance from the x-axis to the line y = 1. Since the line y = 1 is constant, the inner radius does not depend on x.
Step 4: Express the inner radius mathematically. The inner radius is simply the value of y for the inner curve, which is y = 1. Therefore, the inner radius is 1 for all x in the interval [0, 4].
Step 5: Conclude that the inner radius of the washer at any point x in [0, 4] is constant and equal to 1. This is because the inner curve is a horizontal line at y = 1.

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Solid of Revolution

A solid of revolution is a three-dimensional shape created by rotating a two-dimensional area around an axis. In this case, the region R is revolved around the x-axis, forming a solid. Understanding this concept is crucial for visualizing the shape and determining its volume using integration techniques.
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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 cross-sections are washers. It involves subtracting the volume of the inner solid from the outer solid. The inner radius and outer radius are essential for setting up the integral that calculates the volume of the solid formed by revolving the region around the axis.
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Euler's Method

Inner Radius

The inner radius in the context of the washer method refers to the distance from the axis of rotation to the inner curve of the region being revolved. For the given problem, it is the distance from the x-axis to the curve defined by y=1. Identifying this radius is necessary to accurately compute the volume of the solid formed by the revolution.
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Radius of Convergence
Pratica correlata
Domanda del libro di testo

Let R be the region in the first quadrant bounded above by the curve y=2−x² and bounded below by the line y=x. Suppose the shell method is used to determine the volume of the solid generated by revolving R about the y-axis.

b. What is the height of a cylindrical shell at a point x in [0, 2]?

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Domanda del libro di testo

Let R be the region bounded by the curve y=cos^−1x and the x-axis on [0, 1]. A solid of revolution is obtained by revolving R about the y-axis (see figures). 


b. Find an expression for the area A(y) of a cross section of the solid at a point y in [0,π/2]. 

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Domanda del libro di testo

Consider the following curves on the given intervals.  


b. Use a calculator or software to approximate the surface area.


y=cos x, for 0≤x≤π/2; about the x-axis

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Domanda del libro di testo

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


b. The volume of a hemisphere can be computed using the disk method. "

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Domanda del libro di testo

21–30. {Use of Tech} Arc length by calculator


b. If necessary, use technology to evaluate or approximate the integral.

y = cos 2x, for 0 ≤ x ≤ π

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Domanda del libro di testo

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.


b. Suppose P=10, A=20, and r=0. If the initial population is N(0)=100, does the population ever become extinct? Explain.

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