Mathematics
Solid of Revolution
A solid of revolution is a three-dimensional shape that is created by rotating a two-dimensional shape around an axis. This is a common technique used in calculus to find the volume of irregularly shaped objects. The resulting solid can be a cylinder, cone, sphere, or other shapes depending on the original two-dimensional shape and the axis of rotation.
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eBook - PDF- Ron Larson, Bruce Edwards(Authors)
- 2018(Publication Date)
- Cengage Learning EMEA(Publisher)
Solids of revolution are used commonly in engineering and manufacturing. Some examples are axles, funnels, pills, bottles, and pistons, as shown in Figure 7.12. Solids of revolution Figure 7.12 When a region in the plane is revolved about a line, the resulting solid is a Solid of Revolution, and the line is called the axis of revolution. The simplest such solid is a right circular cylinder or disk, which is formed by revolving a rectangle about an axis adjacent to one side of the rectangle, as shown in Figure 7.13. The volume of such a disk is Volume of disk = (area of disk)(width of disk) = π R 2 w where R is the radius of the disk and w is the width. To see how to use the volume of a disk to find the volume of a general Solid of Revolution, consider a Solid of Revolution formed by revolving the plane region in Figure 7.14 (see next page) about the indicated axis. To determine the volume of this solid, consider a representative rectangle in the plane region. When this rectangle is revolved about the axis of revolution, it generates a representative disk whose volume is ΔV = π R 2 Δx. Approximating the volume of the solid by n such disks of width Δx and radius R(x i ) produces Volume of solid ≈ ∑ n i =1 π [R(x i )] 2 Δx = π ∑ n i =1 [R(x i )] 2 Δx. R Rectangle Axis of revolution w R Disk w Volume of a disk: πR 2 w Figure 7.13 7 .2 Volume: The Disk Method 455 R uni0394x x = b x = a Plane region Representative rectangle Solid of Revolution ve Axis of revolution uni0394x Approximation by n disks Representati disk Disk method Figure 7.14 This approximation appears to become better and better as H20648ΔH20648 uni2192 0 (n uni2192 ∞ ). So, you can define the volume of the solid as Volume of solid = lim H20648ΔH20648uni21920 π ∑ n i =1 [R(x i )] 2 Δx = π integral.alt1 b a [R(x)] 2 dx.
eBook - PDFCalculus
Single Variable
- Howard Anton, Irl C. Bivens, Stephen Davis(Authors)
- 2022(Publication Date)
- Wiley(Publisher)
Solution As illustrated in Figure 5.2.7a, we introduce a rectangular coordinate system in which the y-axis passes through the apex and is perpendicular to the base, and the x-axis passes through the base and is parallel to a side of the base. At any y in the interval [0, h] on the y-axis, the cross section perpendicular to the y-axis is a square. If s denotes the length of a side of this square, then by similar triangles (Figure 5.2.7b) 1 2 s 1 2 a = h − y h or s = a h (h − y) Thus, the area A(y) of the cross section at y is A(y) = s 2 = a 2 h 2 (h − y) 2 and by (4) the volume is V = h 0 A(y) dy = h 0 a 2 h 2 (h − y) 2 dy = a 2 h 2 h 0 (h − y) 2 dy = a 2 h 2 − 1 3 (h − y) 3 h y=0 = a 2 h 2 0 + 1 3 h 3 = 1 3 a 2 h That is, the volume is 1 3 of the area of the base times the altitude. Solids of Revolution A Solid of Revolution is a solid that is generated by revolving a plane region about a line that lies in the same plane as the region; the line is called the axis of revolution. Many familiar solids are of this type (Figure 5.2.8). Right circular cylinder Solid sphere Solid cone Hollowed right circular cylinder (a) (b) (c) (d ) Some Familiar Solids of Revolution Axis of revolution FIGURE 5.2.8 Volumes by Disks Perpendicular to the x-axis We will be interested in the following general problem. 284 CHAPTER 5 Applications of the Definite Integral in Geometry, Science, and Engineering 5.2.4 Problem Let f be continuous and nonnegative on [a, b], and let R be the region that is bounded above by y = f (x), below by the x-axis, and on the sides by the lines x = a and x = b (Figure 5.2.9a). Find the volume of the Solid of Revolution that is generated by revolving the region R about the x-axis. x y a b R y = f ( x) (a) f ( x) x y a b x (b) FIGURE 5.2.9 We can solve this problem by slicing. For this purpose, observe that the cross section of the solid taken perpendicular to the x-axis at the point x is a circular disk of radius f (x) (Figure 5.2.9b).
No longer available |Learn more- Daniel C. Alexander, Geralyn M. Koeberlein, , , Daniel C. Alexander, Geralyn M. Koeberlein(Authors)
- 2014(Publication Date)
- Cengage Learning EMEA(Publisher)
Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. EXAMPLE 7 Describe the Solid of Revolution that is formed when a semicircular region having a vertical diameter of length 12 cm [see Figure 9.46(a)] is revolved about that diameter. Then find the exact volume of the solid formed [see Figure 9.46(b)]. SOLUTION The solid that is formed is a sphere with length of radius . The formula we use to find the volume is . Then , which simplifies to . When a circular region is revolved about a line in the circle’s exterior, a doughnut-shaped solid results. The formal name of the resulting Solid of Revolution, shown in Figure 9.47, is the torus . Methods of calculus are necessary to calculate both the surface area and the volume of the torus. Figure 9.47 V 288 p cm 3 V 4 3 p 6 3 V 4 3 p r 3 r 6 cm 9.4 ■ Polyhedrons and Spheres 425 Unless otherwise noted, all content on this page is © Cengage Learning. (a) 12 cm (b) Figure 9.46 EXS. 14–16 1. Which of these two polyhedrons is concave? Note that the interior dihedral angle formed by the planes containing and is larger than 180°. 2. For Figure (a) of Exercise 1, find the number of faces, vertices, and edges in the polyhedron. Then verify Euler’s equation for that polyhedron. 3. For Figure (b) of Exercise 1, find the number of faces, vertices, and edges in the polyhedron. Then verify Euler’s equation for that polyhedron. 4. For a regular tetrahedron, find the number of faces, vertices, and edges in the polyhedron. Then verify Euler’s equation for that polyhedron. 5. For a regular hexahedron, find the number of faces, vertices, and edges in the polyhedron. Then verify Euler’s equation for that polyhedron. 6.
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