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 Surface Area - Set Up Integrals · 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 Surface Area - Set Up Integrals · 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 Surface Area - Set Up Integrals · 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 Surface Area - Set Up Integrals · 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 Surface Area - Set Up Integrals · 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 Surface Area - Set Up Integrals · 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 Surface Area - Exact x-axis · 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 Surface Area - Exact x-axis · 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 Surface Area - Exact x-axis · 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 Surface Area - Exact x-axis · 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 Surface Area - Exact x-axis · 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 Surface Area - Exact x-axis · 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 Surface Area - Exact x-axis · 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 Surface Area - Exact x-axis · 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 Surface Area - About y-axis · 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 Surface Area - About y-axis · 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 Surface Area - About y-axis · 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 Surface Area - About y-axis · 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 Surface Area - Simpson's Rule · 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 Surface Area - Simpson's Rule · 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 Surface Area - Simpson's Rule · 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 Surface Area - Simpson's Rule · 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 Surface Area - CAS x-axis · 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 Surface Area - CAS x-axis · 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 Surface Area - CAS y-axis · 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 Surface Area - CAS y-axis · 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 Surface Area - Gabriel's Horn · 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 Surface Area - Infinite Curve · 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 Surface Area - Loop · 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 Surface Area - Satellite Dish · 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 Surface Area - Ellipsoid · 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 Surface Area - Torus · Level 4
Find the surface area of the torus in Exercise 6.2.63.
33 Surface Area - About Horizontal Line · 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 Surface Area - About y = 4 · 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 Surface Area - Circle About Line · 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 Surface Area - Sphere Zone · 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 Surface Area = Volume · 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 Surface Area - Vertical Shift · 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 Surface Area - General Formula · 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 Example - Surface Area of Sphere Portion · 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 Example - Surface Area Parabola About y-axis · 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 Example - Surface Area Exponential · 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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