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Chapter 8: Problem 115

Let \(y=f(x)\) be positive and strictly increasing on the interval \(0

Short Answer

Expert verified
The disk method and shell method both yield the volume \(V = 2 \pi \int_{a}^{b} x \cdot f(x) dx\) for describing the same revolution, thus proving they yield the same volume.

Step by step solution

01

Disk Method

First, calculate the volume using the disk method. In this method, the differential element is a 'disk' perpendicular to the axis of revolution. The volume of the solid of revolution around the y-axis is given by the integral: \(V = 2 \pi \int_{a}^{b} x \cdot f(x) dx \)
02

Shell Method

Now, calculate the volume using the shell method. This method looks at the differential element as a 'shell'. The volume of the solid of revolution around the y-axis is given by the integral: \(V = 2 \pi \int_{a}^{b} x \cdot f' (x) dx \)
03

Show Equivalence

Since \(f(x)\) is given to be a positive and strictly increasing function, it implies that \(f'(x) > 0\). Hence, the term \(x \cdot f' (x)\) can be substituted with \( d (x \cdot f(x)) \) without changing the value. Hence, the integral in the shell method simplifies to the integral in the disk method, showing that indeed the same volume is obtained with both methods

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Disk Method
The disk method is a technique used in calculus to find the volume of a solid of revolution. This method is most effective when the solid is formed by rotating a region around an axis. Imagine slicing the solid perpendicularly to the axis of rotation, creating circular disks or washers.

For each infinitely thin disk, we calculate its volume and then integrate over the interval to find the total volume.
  • Rotate around the y-axis: The cross-sectional disks have a radius that changes with the height of the cylinder, typically represented by a function of y.
  • The formula for the volume of each disk is: \( V = \pi \int_{c}^{d} [g(y)]^2 \, dy \)where \( g(y) \) is the radius as a function of y.

The method identifies a representation using horizontal lines for disk cross-sections, accumulating every small volume portion across the specified range. This technique is perfect for functions easily expressed as one solid when revolved, and changes with vertical slicing.
Shell Method
The shell method is an alternative calculus technique for determining the volume of a solid of revolution. Instead of slicing perpendicular to the axis, this method visualizes the solid as being composed of cylindrical shells. This approach is especially beneficial when the function is more naturally visualized with vertical slicing.
  • Setting up for rotation around the y-axis means visualizing concentric cylindrical layers or 'shells'. The typical radius of these shells is a function of x.
  • The formula is: \( V = 2 \pi \int_{a}^{b} x \, f(x) \, dx \)where \( x \) is the radius and \( f(x) \) gives the height of the shell.

The shell method provides a natural solution when the axis of rotation is parallel to the axis represented on the graph horizontally, especially if revolving around the y-axis. The approach is often used when dealing with irregular shapes difficult to evaluate using standard disks.
Solid of Revolution
A solid of revolution is a three-dimensional figure obtained by rotating a two-dimensional area around an axis. This process creates a symmetrical solid with circular cross-sections perpendicular to the axis of rotation.
  • The creation: Take any region defined by one or more functions. Revolve this region around a line (the axis) to form the solid.
  • Benefits: This method provides a way to envision and compute the volume of physical shapes formed by symmetrical rotation, like vases or lampshades.

Understanding a solid of revolution is crucial to applying both the disk and shell methods effectively. Whether choosing slices or shells, the concept helps connect the geometry of the 2D region with the rotationally-symmetrical 3D shape.

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Most popular questions from this chapter

The Gamma Function \(\Gamma(n)\) is defined by $$ \Gamma(n)=\int_{0}^{\infty} x^{n-1} e^{-x} d x, \quad n>0. $$ (a) Find \(\Gamma(1), \Gamma(2)\), and \(\Gamma(3)\). (b) Use integration by parts to show that \(\Gamma(n+1)=n \Gamma(n)\). (c) Write \(\Gamma(n)\) using factorial notation where \(n\) is a positive integer.

Use the integration capabilities of a graphing utility to approximate the are length of the curve over the given interval. $$ y=x^{2 / 3}, \quad[1,8] $$

Show that \(\lim _{x \rightarrow \infty} \frac{x^{n}}{e^{x}}=0\) for any integer \(n>0\).

Consider the integral \(\int_{0}^{\pi / 2} \frac{4}{1+(\tan x)^{n}} d x\) where \(n\) is a positive integer. (a) Is the integral improper? Explain. (b) Use a graphing utility to graph the integrand for \(n=2,4\), 8 , and 12 . (c) Use the graphs to approximate the integral as \(n \rightarrow \infty\). (d) Use a computer algebra system to evaluate the integral for the values of \(n\) in part (b). Make a conjecture about the value of the integral for any positive integer \(n .\) Compare your results with your answer in part (c).

Laplace Transforms Let \(f(t)\) be a function defined for all positive values of \(t\). The Laplace Transform of \(f(t)\) is defined by$$F(s)=\int_{0}^{\infty} e^{-s t} f(t) d t$$if the improper integral exists. Laplace Transforms are used to solve differential equations, find the Laplace Transform of the function. $$ f(t)=\cosh a t $$
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