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Chapter 7: Problem 20

In Exercises 13-22, use the shell method to write and evaluate the definite integral that represents the volume of the solid generated by revolving the plane region about the x-axis. $$y=3-x, y=0, x=6$$

Short Answer

Expert verified
The volume of the solid generated by revolving the plane region bounded by \(y=0\), \(y=3-x\), and \(x=6\) about the x-axis is \(36 \pi\) cubic units.

Step by step solution

01

Identify the revolving figure

Sketch and identify the bounded region. The bounded region is formed by the x-axis, y-axis, the line represented by \(y=3-x\), and the vertical line \(x=6\). This region takes a triangular shape and will be revolved around the x-axis.
02

Setting up the integral for volume

By using the shell method, the volume V of the solid is given by the definite integral: \(V = 2 \pi \int_{a}^{b} p(h)h' dh\), where \(p(h)\) is the radius function and \(h'\) is the height function. Here, the integral limits are from \(a=0\) to \(b=6\). The radius function \(p(h)\) equals \(h\), as the height from the x-axis (where the figure is revolving) to the curve \(y = 3 - h\). The height function \(h'\) is the height of the cylindrical shells, which equals to \(3 - h\). Plug these into the volume formula.
03

Evaluating the integral to find the volume

The volume of the solid is then given by the definite integral: \(V = 2 \pi \int_{0}^{6} h(3 - h) dh\) = \(2 \pi [ \frac{3h虏}{2} - \frac{h鲁}{3}]_{0}^{6}\), which simplifies to \(2 \pi [ 54 - 72 ] = -2 \pi * 18\). The volume cannot be negative, so take the absolute value to get \(36 \pi\).
04

Finalize the volume of the solid

Therefore, the volume of the solid generated by rotating the bounded region around the x-axis is \(36 \pi\) cubic units. The negative sign that arose was merely a computational sign, in practice, volume is always stated in positive units. Hence, the absolute value was taken to reflect this.

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Key Concepts

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

Definite Integral
A definite integral represents the accumulation of quantities, which, in the context of geometric shapes, usually corresponds to an area, volume, or length. When calculating using definite integrals, we consider a function defined over a certain interval and take the limit of a sum of product terms to get a precise accumulation.

For the shell method, we essentially convert a region (in this case, under a line) into an infinite sum of small cylindrical shells. We sum up their respective volumes by integrating over a given interval.
  • The lower limit of integration represents the start of the region of interest.
  • The upper limit marks the end of that region.
Integrating over these limits provides an exact value for the volume enclosed by the cylindrical shells.
Volume of Solids
The volume of solids is a three-dimensional space enclosed within a solid object. Finding the volume involves summing across the width, depth, and height of the object.

There are several methods to determine the volume, one of which involves revolving a region around an axis to form a solid. This method is ideal for consistency with symmetrical objects like cylinders or spheres and is crucial in disciplines like physics and engineering.
  • Use calculus methods, such as integration, to sum small elements.
  • Visualize solids as a series of layers or shells, each contributing to the total volume.
By picturing these solids as made of infinitesimally small sections, it becomes possible to compute their volume precisely using mathematical techniques like integration.
Revolving Around the x-axis
Revolving a region around the x-axis involves creating a three-dimensional shape or solid by spinning a two-dimensional area around this axis.

This method is particularly effective with circular or symmetrical shapes, which easily convert into volumes via this approach. It's essential to grasp this concept when using the shell method to compute volumes.
  • The axis of revolution determines the shape's symmetry.
  • The revolving effect turns simple shapes into intricate volumetric solids.
This concept helps solve problems involving real-world objects and phenomena, which require measuring or estimating volumes accurately.
Cylindrical Shells Method
The cylindrical shells method is a technique for finding the volume of a solid of revolution. Unlike the disk or washer methods, this technique revolves a region around an axis parallel to itself, creating a series of cylindrical shells.

This method uses the idea that you can view a shape as composed of consecutive cylindrical shells. The formula for the volume involves a definite integral which sums the lateral surface areas of these shells to find the total volume:
  • Identify the height and radius of each shell segment.
  • Integrate the resulting function over the interval of the bounded region.
  • Multiply by the constant factor, typically involving \(2 \pi\), representing the circumference of the shell's base.
The cylindrical shells method is particularly useful when the axis of revolution is not perpendicular to the axis of the function. This versatility makes it ideal for diverse application scenarios.

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Most popular questions from this chapter

$$\begin{array}{l}{\text { Equal Volumes Let } V_{1} \text { and } V_{2} \text { be the volumes of the }} \\ {\text { solids that result when the plane region bounded by }} \\ {y=\frac{1}{x^{\prime}}, y=0, \quad x=\frac{1}{4}, \quad \text { and } \quad x=c, \quad c>\frac{1}{4}}\end{array}$$ $$\begin{array}{l}{\text { is revolved about the } x \text { -axis and the } y \text { -axis, respectively. Find }} \\ {\text { the value of } c \text { for which } V_{1}=V_{2}}\end{array}$$

Boyle's Law In Exercises 43 and 44, find the work done by the gas for the given volume and pressure. Assume that the pressure is inversely proportional to the volume. (See Example 6.) A quantity of gas with an initial volume of 2 cubic feet and a pressure of 1000 pounds per square foot expands to a volume of 3 cubic feet.

Dividing a Solid In Exercises 57 and 58 , consider the solid formed by revolving the region bounded by \(y=\sqrt{x}, y=0,\) \(x=1,\) and \(x=3\) about the \(x\) -axis. Find the value of \(x\) in the interval \([1,3]\) that divides the solid into two parts of equal volume.

Hydraulic Press In Exercises \(45-48,\) use the integration capabilities of a graphing utility to approximate the work done by a press in a manufacturing process. A model for the variable force \(F\) (in pounds) and the distance \(x\) (in feet) the press moves is given. $$F(x)=1000 \sinh x \quad 0 \leq x \leq 2$$

Approximation In Exercises 51 and \(52,\) determine which value best approximates the length of the arc represented by the integral. Make your selection on the basis of a sketch of the arc, not by performing calculations. \(\int_{0}^{\pi / 4} \sqrt{1+\left[\frac{d}{d x}(\tan x)\right]^{2}} d x\) (a) 3\(\quad\) (b) \(-2 \quad\) (c) 4\(\quad\) (d) \(\frac{4 \pi}{3}\) (e) 1
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