Surface area of a torus When the circle x2 + 1y - a22 = r2

What is the area of the curved surface of a right circular cone of radius 3 and height 4?

A frustum of a cone is generated by revolving the graph of y = 4x on the interval 32, 64 about the x-axis. What is the area of the surface of the frustum?

Suppose f is positive and differentiable on 3a, b4. The curve y = f 1x2 on 3a, b4 is revolved about the x-axis. Explain how to find the area of the surface that is generated.

Suppose g is positive and differentiable on 3c, d4. The curve x = g1y2 on 3c, d4 is revolved about the y-axis. Explain how to find the area of the surface that is generated. Basic Skills

514. Computing surface areas Find the area of the surface generated when the given curve is revolved about the x-axis. y = 3x + 4 on 30, 64

514. Computing surface areas Find the area of the surface generated when the given curve is revolved about the x-axis. y = 12 - 3x on 31, 34

514. Computing surface areas Find the area of the surface generated when the given curve is revolved about the x-axis. y = 82x on 39, 204

514. Computing surface areas Find the area of the surface generated when the given curve is revolved about the x-axis. y = x3 on 30, 14

514. Computing surface areas Find the area of the surface generated when the given curve is revolved about the x-axis. y = x3>2 - x1>2 3 on 31, 24

514. Computing surface areas Find the area of the surface generated when the given curve is revolved about the x-axis. y = 24x + 6 on 30, 54

514. Computing surface areas Find the area of the surface generated when the given curve is revolved about the x-axis. y = 1 4 1e2x + e-2x2 on 3-2, 24

514. Computing surface areas Find the area of the surface generated when the given curve is revolved about the x-axis. y = x4 8 + 1 4x2 on 31, 24

514. Computing surface areas Find the area of the surface generated when the given curve is revolved about the x-axis. y = x3 3 + 1 4x on c 1 2 , 2 d

514. Computing surface areas Find the area of the surface generated when the given curve is revolved about the x-axis. y = 25x - x2 on 31, 44

1516. Painting surfaces A 1.5-mm layer of paint is applied to one side of the following surfaces. Find the approximate volume of paint needed. Assume that x and y are measured in meters. The spherical zone generated when the curve y = 28x - x2 on the interval 31, 74 is revolved about the x-axis

1516. Painting surfaces A 1.5-mm layer of paint is applied to one side of the following surfaces. Find the approximate volume of paint needed. Assume that x and y are measured in meters. The spherical zone generated when the upper portion of the circle x2 + y2 = 100 on the interval 3-8, 84 is revolved about the x-axis

1720. Revolving about the y-axis Find the area of the surface generated when the given curve is revolved about the y-axis. y = 13x21>3, for 0 x 8 3

1720. Revolving about the y-axis Find the area of the surface generated when the given curve is revolved about the y-axis. y = x2 4 , for 2 x 4

1720. Revolving about the y-axis Find the area of the surface generated when the given curve is revolved about the y-axis. The part of the curve y = 4x - 1 between the points 11, 32 and 14, 152

1720. Revolving about the y-axis Find the area of the surface generated when the given curve is revolved about the y-axis. The part of the curve y = 12 ln 12x + 24x2 - 12 between the points 112 , 02 and 117 16, ln 22 Further Explorations

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. If the curve y = f 1x2 on the interval 3a, b4 is revolved about the y-axis, the area of the surface generated is L f 1b2 f 1a2 2p f 1y221 + f _1y22 dy. b. If f is not one-to-one on the interval 3a, b4, then the area of the surface generated when the graph of f on 3a, b4 is revolved about the x-axis is not defined. c. Let f 1x2 = 12x2. The area of the surface generated when the graph of f on 3-4, 44 is revolved about the x-axis is twice the area of the surface generated when the graph of f on 30, 44 is revolved about the x-axis. d. Let f 1x2 = 12x2. The area of the surface generated when the graph of f on 3-4, 44 is revolved about the y-axis is twice the area of the surface generated when the graph of f on 30, 44 is revolved about the y-axis.

2225. Surface area calculations Use the method of your choice to determine the area of the surface generated when the following curves are revolved about the indicated axis. x = 212y - y2, for 2 y 10; about the y-axis

2225. Surface area calculations Use the method of your choice to determine the area of the surface generated when the following curves are revolved about the indicated axis. x = 4y3>2 - y1>2 12 , for 1 y 4; about the y-axis

2225. Surface area calculations Use the method of your choice to determine the area of the surface generated when the following curves are revolved about the indicated axis. y = 1 + 21 - x2 between the points 11, 12 and a 23 2 , 3 2 b; about the y-axis

2225. Surface area calculations Use the method of your choice to determine the area of the surface generated when the following curves are revolved about the indicated axis. y = 9x2>3 - x4>3 32 , for 1 x 8; about the x-axis

2629. Surface area using technology Consider the following curves on the given intervals. a. Write the integral that gives the area of the surface generated when the curve is revolved about the x-axis. b. Use a calculator or software to approximate the surface area. y = x5 on 30, 14

2629. Surface area using technology Consider the following curves on the given intervals. a. Write the integral that gives the area of the surface generated when the curve is revolved about the x-axis. b. Use a calculator or software to approximate the surface area. y = cos x on c 0, p 2 d

2629. Surface area using technology Consider the following curves on the given intervals. a. Write the integral that gives the area of the surface generated when the curve is revolved about the x-axis. b. Use a calculator or software to approximate the surface area. y = ln x2 on 31, 1e4

2629. Surface area using technology Consider the following curves on the given intervals. a. Write the integral that gives the area of the surface generated when the curve is revolved about the x-axis. b. Use a calculator or software to approximate the surface area. y = tan x on c 0, p 4 d

Cones and cylinders The volume of a cone of radius r and height h is one-third the volume of a cylinder with the same radius and height. Does the surface area of a cone of radius r and height h equal one-third the surface area of a cylinder with the same radius and height? If not, find the correct relationship. Exclude the bases of the cone and cylinder.

Revolving an astroid Consider the upper half of the astroid described by x2>3 + y2>3 = a2>3, where a 7 0 and _ x _ a. Find the area of the surface generated when this curve is revolved about the x-axis. Use symmetry. Note that the function describing the curve is not differentiable at 0. However, the surface area integral can be evaluated using methods you know. Applications

Surface area of a torus When the circle x2 + 1y - a22 = r2 on the interval 3-r, r4 is revolved about the x-axis, the result is the surface of a torus, where 0 6 r 6 a. Show that the surface area of the torus is S = 4p2ar.

Zones of a sphere Suppose a sphere of radius r is sliced by two horizontal planes h units apart (see figure). Show that the surface area of the resulting zone on the sphere is 2prh, independent of the location of the cutting planes.

Surface area of an ellipsoid If the top half of the ellipse x2 a2 + y2 b2 = 1 is revolved about the x-axis, the result is an ellipsoid whose axis along the x-axis has length 2a, whose axis along the y-axis has length 2b, and whose axis perpendicular to the xy-plane has length 2b. We assume that 0 6 b 6 a (see figure). Use the following steps to find the surface area S of this ellipsoid. a x b y _ _ 1 x2 a2 y2 b2 a. Use the surface area formula to show that S = 4pb a L a 0 2a2 - c2x2 dx, where c2 = 1 - b2 a2. b. Use the change of variables u = cx to show that S = 4pb 2a2 - b2L 2a2 - b2 0 2a2 - u2 du. c. A table of integrals shows that L 2a2 - u2 du = 1 2 au2a2 - u2 + a2 sin-1 u a b + C. Use this fact to show that the surface area of the ellipsoid is S = 2pbab + a2 2a2 - b2 sin-1 2a2 - b2 a b. d. If a and b have units of length (say, meters), what are the units of S according to this formula? e. Use part (a) to show that if a = b, then S = 4pa2, which is the surface area of a sphere of radius a.

Surface-area-to-volume ratio (SAV) In the design of solid objects (both artificial and natural), the ratio of the surface area to the volume of the object is important. Animals typically generate heat at a rate proportional to their volume and lose heat at a rate proportional to their surface area. Therefore, animals with a low SAV ratio tend to retain heat, whereas animals with a high SAV ratio (such as children and hummingbirds) lose heat relatively quickly. a. What is the SAV ratio of a cube with side lengths a? b. What is the SAV ratio of a ball with radius a? c. Use the result of Exercise 34 to find the SAV ratio of an ellipsoid whose long axis has length 2a13 4, for a 1, and whose other two axes have half the length of the long axis. (This scaling is used so that, for a given value of a, the volumes of the ellipsoid and the ball of radius a are equal.) The volume of a general ellipsoid is V = 4p 3 ABC, where the axes have lengths 2A, 2B, and 2C. d. Graph the SAV ratio of the ball of radius a 1 as a function of a (part (b)) and graph the SAV ratio of the ellipsoid described in (part (c)) on the same set of axes. Which object has the smaller SAV ratio? e. Among all ellipsoids of a fixed volume, which one would you choose for the shape of an animal if the goal is to minimize heat loss? Additional Exercises

Surface area of a frustum Show that the surface area of the frustum of a cone generated by revolving the line segment between 1a, g1a22 and 1b, g1b22 about the x-axis is p1g1b2 + g1a22/, for any linear function g1x2 = cx + d that is positive on the interval 3a, b4, where / is the slant height of the frustum.

Scaling surface area Let f be a nonnegative function with a continuous first derivative on 3a, b4 and suppose that g1x2 = c f 1x2 and h1x2 = f 1cx2, where c 7 0. When the curve y = f 1x2 on 3a, b4 is revolved about the x-axis, the area of the resulting surface is A. Evaluate the following integrals in terms of A and c. a. L b a 2pg1x22c2 + g_1x22 dx b. L b>c a>c 2ph1x22c2 + h_1x22 dx

Surface plus cylinder Suppose f is a nonnegative function with a continuous first derivative on 3a, b4. Let L equal the length of the graph of f on 3a, b4 and let S be the area of the surface generated by revolving the graph of f on 3a, b4 about the x-axis. For a positive constant C, assume the curve y = f 1x2 + C is revolved about the x-axis. Show that the area of the resulting surface equals the sum of S and the surface area of a right circular cylinder of radius C and height L.