Stewart 8th Section 8.2: Area of a Surface of Revolution

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Stewart 8th Section 8.2: Area of a Surface of Revolution 0/42
1 Applications of Integration · Level 2
(a) Set up an integral for the area of the surface obtained by rotating the curve \(y = \tan x\), \(0 \leq x \leq \dfrac{\pi}{3}\), about (i) the \(x\)-axis and (ii) the \(y\)-axis. (b) Use the numerical integration capability of a calculator to evaluate the surface areas correct to four decimal places.
2 Applications of Integration · Level 2
(a) Set up an integral for the area of the surface obtained by rotating the curve \(y = x^{-2}\), \(1 \leq x \leq 2\), about (i) the \(x\)-axis and (ii) the \(y\)-axis. (b) Use the numerical integration capability of a calculator to evaluate the surface areas correct to four decimal places.
3 Applications of Integration · Level 2
(a) Set up an integral for the area of the surface obtained by rotating the curve \(y = e^{-x^2}\), \(-1 \leq x \leq 1\), about (i) the \(x\)-axis and (ii) the \(y\)-axis. (b) Use the numerical integration capability of a calculator to evaluate the surface areas correct to four decimal places.
4 Applications of Integration · Level 2
(a) Set up an integral for the area of the surface obtained by rotating the curve \(x = \ln(2 y + 1)\), \(0 \leq y \leq 1\), about (i) the \(x\)-axis and (ii) the \(y\)-axis. (b) Use the numerical integration capability of a calculator to evaluate the surface areas correct to four decimal places.
5 Applications of Integration · Level 2
(a) Set up an integral for the area of the surface obtained by rotating the curve \(x = y + y^3\), \(0 \leq y \leq 1\), about (i) the \(x\)-axis and (ii) the \(y\)-axis. (b) Use the numerical integration capability of a calculator to evaluate the surface areas correct to four decimal places.
6 Applications of Integration · Level 2
(a) Set up an integral for the area of the surface obtained by rotating the curve \(y = \tan^{-1} x\), \(0 \leq x \leq 2\), about (i) the \(x\)-axis and (ii) the \(y\)-axis. (b) Use the numerical integration capability of a calculator to evaluate the surface areas correct to four decimal places.
7 Applications of Integration · Level 2
Find the exact area of the surface obtained by rotating the curve \(y = x^3\), \(0 \leq x \leq 2\), about the \(x\)-axis.
8 Applications of Integration · Level 2
Find the exact area of the surface obtained by rotating the curve \(y = \sqrt{5 - x}\), \(3 \leq x \leq 5\), about the \(x\)-axis.
9 Applications of Integration · Level 2
Find the exact area of the surface obtained by rotating the curve \(y^2 = x + 1\), \(0 \leq x \leq 3\), about the \(x\)-axis.
10 Applications of Integration · Level 3
Find the exact area of the surface obtained by rotating the curve \(y = \sqrt{1 + e^x}\), \(0 \leq x \leq 1\), about the \(x\)-axis.
11 Applications of Integration · Level 2
Find the exact area of the surface obtained by rotating the curve \(y = \cos\left(\dfrac{1}{2} x\right)\), \(0 \leq x \leq \pi\), about the \(x\)-axis.
12 Applications of Integration · Level 3
Find the exact area of the surface obtained by rotating the curve \(y = \dfrac{x^3}{6} + \dfrac{1}{2 x}\), \(\dfrac{1}{2} \leq x \leq 1\), about the \(x\)-axis.
13 Applications of Integration · Level 3
Find the exact area of the surface obtained by rotating the curve \(x = \dfrac{1}{3}(y^2 + 2)^{\dfrac{3}{2}}\), \(1 \leq y \leq 2\), about the \(x\)-axis.
14 Applications of Integration · Level 2
Find the exact area of the surface obtained by rotating the curve \(x = 1 + 2 y^2\), \(1 \leq y \leq 2\), about the \(x\)-axis.
15 Applications of Integration · Level 2
The given curve is rotated about the \(y\)-axis. Find the area of the resulting surface: \(y = \dfrac{1}{3} x^{\dfrac{3}{2}}\), \(0 \leq x \leq 12\).
16 Applications of Integration · Level 3
The given curve is rotated about the \(y\)-axis. Find the area of the resulting surface: \(x^{\dfrac{2}{3}} + y^{\dfrac{2}{3}} = 1\), \(0 \leq y \leq 1\).
17 Applications of Integration · Level 2
The given curve is rotated about the \(y\)-axis. Find the area of the resulting surface: \(x = \sqrt{a^2 - y^2}\), \(0 \leq y \leq \dfrac{a}{2}\).
18 Applications of Integration · Level 3
The given curve is rotated about the \(y\)-axis. Find the area of the resulting surface: \(y = \dfrac{1}{4} x^2 - \dfrac{1}{2} \ln x\), \(1 \leq x \leq 2\).
19 Applications of Integration · Level 3
Use Simpson's Rule with \(n = 10\) to approximate the area of the surface obtained by rotating the curve \(y = \dfrac{1}{5} x^5\), \(0 \leq x \leq 5\), about the \(x\)-axis. Compare your answer with the value of the integral produced by a calculator.
20 Applications of Integration · Level 3
Use Simpson's Rule with \(n = 10\) to approximate the area of the surface obtained by rotating the curve \(y = x + x^2\), \(0 \leq x \leq 1\), about the \(x\)-axis. Compare your answer with the value of the integral produced by a calculator.
21 Applications of Integration · Level 3
Use Simpson's Rule with \(n = 10\) to approximate the area of the surface obtained by rotating the curve \(y = x e^x\), \(0 \leq x \leq 1\), about the \(x\)-axis. Compare your answer with the value of the integral produced by a calculator.
22 Applications of Integration · Level 3
Use Simpson's Rule with \(n = 10\) to approximate the area of the surface obtained by rotating the curve \(y = x \ln x\), \(1 \leq x \leq 2\), about the \(x\)-axis. Compare your answer with the value of the integral produced by a calculator.
23 Applications of Integration · Level 3
Use either a CAS or a table of integrals to find the exact area of the surface obtained by rotating the curve \(y = \dfrac{1}{x}\), \(1 \leq x \leq 2\), about the \(x\)-axis.
24 Applications of Integration · Level 3
Use either a CAS or a table of integrals to find the exact area of the surface obtained by rotating the curve \(y = \sqrt{x^2 + 1}\), \(0 \leq x \leq 3\), about the \(x\)-axis.
25 Applications of Integration · Level 3
Use a CAS to find the exact area of the surface obtained by rotating the curve \(y = x^3\), \(0 \leq y \leq 1\), about the \(y\)-axis. If your CAS has trouble evaluating the integral, express the surface area as an integral in the other variable.
26 Applications of Integration · Level 3
Use a CAS to find the exact area of the surface obtained by rotating the curve \(y = \ln(x + 1)\), \(0 \leq x \leq 1\), about the \(y\)-axis. If your CAS has trouble evaluating the integral, express the surface area as an integral in the other variable.
27 Applications of Integration · Level 4
If the region \(cal(R) = \{(x, y) | x \geq 1, 0 \leq y \leq \dfrac{1}{x}\}\) is rotated about the \(x\)-axis, the volume of the resulting solid is finite. Show that the surface area is infinite. (The surface is known as Gabriel's horn.)
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28 Applications of Integration · Level 4
If the infinite curve \(y = e^{-x}\), \(x \geq 0\), is rotated about the \(x\)-axis, find the area of the resulting surface.
29 Applications of Integration · Level 4
(a) If \(a > 0\), find the area of the surface generated by rotating the loop of the curve \(3 a y^2 = x(a - x)^2\) about the \(x\)-axis. (b) Find the surface area if the loop is rotated about the \(y\)-axis.
30 Applications of Integration · Level 3
A group of engineers is building a parabolic satellite dish whose shape will be formed by rotating the curve \(y = a x^2\) about the \(y\)-axis. If the dish is to have a 10-ft diameter and a maximum depth of 2 ft, find the value of \(a\) and the surface area of the dish.
31 Applications of Integration · Level 4
(a) The ellipse \(\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1\), \(a > b\), is rotated about the \(x\)-axis to form a surface called an ellipsoid, or prolate spheroid. Find the surface area of this ellipsoid. (b) If the ellipse in part (a) is rotated about its minor axis (the \(y\)-axis), the resulting ellipsoid is called an oblate spheroid. Find the surface area of this ellipsoid.
32 Applications of Integration · Level 4
Find the surface area of the torus in Exercise 6.2.63.
33 Applications of Integration · Level 3
If the curve \(y = f(x)\), \(a \leq x \leq b\), is rotated about the horizontal line \(y = c\), where \(f(x) \leq c\), find a formula for the area of the resulting surface.
34 Applications of Integration · Level 3
Use the result of Exercise 33 to set up an integral to find the area of the surface generated by rotating the curve \(y = \sqrt{x}\), \(0 \leq x \leq 4\), about the line \(y = 4\). Then use a CAS to evaluate the integral.
35 Applications of Integration · Level 4
Find the area of the surface obtained by rotating the circle \(x^2 + y^2 = r^2\) about the line \(y = r\).
36 Applications of Integration · Level 4
(a) Show that the surface area of a zone of a sphere that lies between two parallel planes is \(S = 2 \pi R h\), where \(R\) is the radius of the sphere and \(h\) is the distance between the planes. (Notice that \(S\) depends only on the distance between the planes and not on their location, provided that both planes intersect the sphere.) (b) Show that the surface area of a zone of a cylinder with radius \(R\) and height \(h\) is the same as the surface area of the zone of a sphere in part (a).
37 Applications of Integration · Level 4
Show that if we rotate the curve \(y = e^{\dfrac{x}{2}} + e^{-\dfrac{x}{2}}\) about the \(x\)-axis, the area of the resulting surface is the same value as the enclosed volume for any interval \(a \leq x \leq b\).
38 Applications of Integration · Level 4
Let \(L\) be the length of the curve \(y = f(x)\), \(a \leq x \leq b\), where \(f\) is positive and has a continuous derivative. Let \(S_f\) be the surface area generated by rotating the curve about the \(x\)-axis. If \(c\) is a positive constant, define \(g(x) = f(x) + c\) and let \(S_g\) be the corresponding surface area generated by the curve \(y = g(x)\), \(a \leq x \leq b\). Express \(S_g\) in terms of \(S_f\) and \(L\).
39 Applications of Integration · Level 4
Formula 4 is valid only when \(f(x) \geq 0\). Show that when \(f(x)\) is not necessarily positive, the formula for surface area becomes \(S = \displaystyle\int_{a}^{b} 2 \pi |f(x)| \sqrt{1 + [f'(x)]^2} d x\).
40 Applications of Integration · Level 2
The curve \(y = \sqrt{4 - x^2}\), \(-1 \leq x \leq 1\), is an arc of the circle \(x^2 + y^2 = 4\). Find the area of the surface obtained by rotating this arc about the \(x\)-axis. (The surface is a portion of a sphere of radius 2.)
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41 Applications of Integration · Level 3
The arc of the parabola \(y = x^2\) from \((1, 1)\) to \((2, 4)\) is rotated about the \(y\)-axis. Find the area of the resulting surface.
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42 Applications of Integration · Level 4
Find the area of the surface generated by rotating the curve \(y = e^x\), \(0 \leq x \leq 1\), about the \(x\)-axis.

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