Stewart 8th §6.3: Volumes by Cylindrical Shells

52문제

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Stewart 8th §6.3: Volumes by Cylindrical Shells 0/52
1 Applications of Integration · Level 3
Let \(S\) be the solid obtained by rotating the region shown in the figure about the \(y\)-axis. Explain why it is awkward to use slicing to find the volume \(V\) of \(S\). Sketch a typical approximating shell. What are its circumference and height? Use shells to find \(V\). [Figure: \(y = x(x - 1)^2\)]
2 Applications of Integration · Level 3
Let \(S\) be the solid obtained by rotating the region shown in the figure about the \(y\)-axis. Sketch a typical cylindrical shell and find its circumference and height. Use shells to find the volume of \(S\). Do you think this method is preferable to slicing? Explain. [Figure: \(y = \sin(x^2)\)]
3 Applications of Integration · Level 2
Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the \(y\)-axis. \(y = \sqrt[3]{x}\), \(y = 0\), \(x = 1\)
Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the \(y\)-axis.
4 Applications of Integration · Level 2
(a)\(y = x^3\), \(y = 0\), \(x = 1\), \(x = 2\)
5 Applications of Integration · Level 2
(b)\(y = e^{-x^2}\), \(y = 0\), \(x = 0\), \(x = 1\)
6 Applications of Integration · Level 2
(c)\(y = 4x - x^2\), \(y = x\)
7 Applications of Integration · Level 2
(d)\(y = x^2\), \(y = 6x - 2x^2\)
8 Applications of Integration · Level 3
Let \(V\) be the volume of the solid obtained by rotating about the \(y\)-axis the region bounded by \(y = \sqrt{x}\) and \(y = x^2\). Find \(V\) both by slicing and by cylindrical shells. In both cases draw a diagram to explain your method.
9 Applications of Integration · Level 3
Use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the given curves about the \(x\)-axis. \(x y = 1\), \(x = 0\), \(y = 1\), \(y = 3\)
Use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the given curves about the \(x\)-axis.
10 Applications of Integration · Level 2
(a)\(y = \sqrt{x}\), \(x = 0\), \(y = 2\)
11 Applications of Integration · Level 3
(b)\(y = x^{\frac{3}{2}}\), \(y = 8\), \(x = 0\)
12 Applications of Integration · Level 3
(c)\(x = -3y^2 + 12y - 9\), \(x = 0\)
13 Applications of Integration · Level 2
(d)\(x = 1 + (y - 2)^2\), \(x = 2\)
Set up, but do not evaluate, an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis.
14 Applications of Integration · Level 3
\(x + y = 4\), \(x = y^2 - 4y + 4\)
15 Applications of Integration · Level 3
Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the specified axis. \(y = x^3\), \(y = 8\), \(x = 0\); about \(x = 3\)
16 Applications of Integration · Level 2
\(y = 4 - 2x\), \(y = 0\), \(x = 0\); about \(x = -1\)
17 Applications of Integration · Level 3
\(y = 4x - x^2\), \(y = 3\); about \(x = 1\)
18 Applications of Integration · Level 3
\(y = \sqrt{x}\), \(x = 2y\); about \(x = 5\)
19 Applications of Integration · Level 3
\(x = 2y^2\), \(y \geq 0\), \(x = 2\); about \(y = 2\)
20 Applications of Integration · Level 3
\(x = 2y^2\), \(x = y^2 + 1\); about \(y = -2\)
21 Applications of Integration · Level 3
(a) Set up an integral for the volume of the solid obtained by rotating the region bounded by the given curve about the specified axis.
(b) Use your calculator to evaluate the integral correct to five decimal places. \(y = x e^{-x}\), \(y = 0\), \(x = 2\); about the \(y\)-axis

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22 Applications of Integration · Level 4
\(y = \tan x\), \(y = 0\), \(x = \dfrac{\pi}{4}\); about \(x = \dfrac{\pi}{2}\)
23 Applications of Integration · Level 3
\(y = \cos^4 x\), \(y = -\cos^4 x\), \(-\dfrac{\pi}{2} \leq x \leq \dfrac{\pi}{2}\); about \(x = \pi\)
24 Applications of Integration · Level 4
\(y = x\), \(y = \dfrac{2x}{1 + x^3}\); about \(x = -1\)
25 Applications of Integration · Level 3
\(x = \sqrt{\sin y}\), \(0 \leq y \leq \pi\), \(x = 0\); about \(y = 4\)
26 Applications of Integration · Level 4
\(x^2 - y^2 = 7\), \(x = 4\); about \(y = 5\)
27 Applications of Integration · Level 3
Use the Midpoint Rule with \(n = 5\) to estimate the volume obtained by rotating about the \(y\)-axis the region under the curve \(y = \sqrt{1 + x^3}\), \(0 \leq x \leq 1\).
28 Applications of Integration · Level 3
If the region shown in the figure is rotated about the \(y\)-axis to form a solid, use the Midpoint Rule with \(n = 5\) to estimate the volume of the solid. [Figure with curve from \(x = 0\) to \(x = 10\)]
29 Applications of Integration · Level 2
Each integral represents the volume of a solid. Describe the solid. \(\displaystyle\int_{0}^{3} 2 \pi x^5 d x\)
Each integral represents the volume of a solid. Describe the solid.
30 Applications of Integration · Level 2
(a)\(\displaystyle\int_{1}^{2} 2 \pi y \ln y d y\)
31 Applications of Integration · Level 3
(b)\(2 \pi \displaystyle\int_{1}^{4} \dfrac{y + 2}{y^2} d y\)
32 Applications of Integration · Level 3
(c)\(\displaystyle\int_{0}^{1} 2 \pi (2 - x)(3^x - 2^x) d x\)
33 Applications of Integration · Level 4
Use a graph to estimate the \(x\)-coordinates of the points of intersection of the given curves. Then use your calculator to estimate the volume of the solid obtained by rotating about the \(y\)-axis the region enclosed by these curves. \(y = x^2 - 2x\), \(y = \dfrac{x}{x^2 + 1}\)
Use a graph to estimate the x-coordinates of the points of intersection of the given curves. Then use this information and your calculator to estimate the volume of the solid obtained by rotating about the y-axis the region enclosed by these curves.
34 Applications of Integration · Level 5
\(y = e^{\sin x}\), \(y = x^2 - 4x + 5\)
35 Applications of Integration · Level 4
Use a computer algebra system to find the exact volume of the solid obtained by rotating the region bounded by the given curves about the specified line. \(y = \sin^2 x\), \(y = \sin^4 x\), \(0 \leq x \leq \pi\); about \(x = \dfrac{\pi}{2}\)
36 Applications of Integration · Level 5
\(y = x^3 \sin x\), \(y = 0\), \(0 \leq x \leq \pi\); about \(x = -1\)
37 Applications of Integration · Level 3
The region bounded by the given curves is rotated about the specified axis. Find the volume of the resulting solid by any method. \(y = -x^2 + 6x - 8\), \(y = 0\); about the \(y\)-axis
38 Applications of Integration · Level 2
\(y = -x^2 + 6x - 8\), \(y = 0\); about the \(x\)-axis
39 Applications of Integration · Level 3
\(y^2 - x^2 = 1\), \(y = 2\); about the \(x\)-axis
40 Applications of Integration · Level 3
\(y^2 - x^2 = 1\), \(y = 2\); about the \(y\)-axis
41 Applications of Integration · Level 3
\(x^2 + (y - 1)^2 = 1\); about the \(y\)-axis
42 Applications of Integration · Level 3
\(x = (y - 3)^2\), \(x = 4\); about \(y = 1\)
43 Applications of Integration · Level 4
\(x = (y - 1)^2\), \(x - y = 1\); about \(x = -1\)
44 Applications of Integration · Level 4
Let \(T\) be the triangular region with vertices \((0, 0)\), \((1, 0)\), and \((1, 2)\), and let \(V\) be the volume of the solid generated when \(T\) is rotated about the line \(x = a\), where \(a > 1\). Express \(a\) in terms of \(V\).
45 Applications of Integration · Level 3
Use cylindrical shells to find the volume of the solid.
A sphere of radius \(r\).
46 Applications of Integration · Level 4
The solid torus of Exercise 6.2.63.
47 Applications of Integration · Level 3
A right circular cone with height \(h\) and base radius \(r\).
48 Applications of Integration · Level 4
Suppose you make napkin rings by drilling holes with different diameters through two wooden balls (which also have different diameters). You discover that both napkin rings have the same height \(h\), as shown in the figure.
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(a) Guess which ring has more wood in it.
(b) Check your guess: Use cylindrical shells to compute the volume of a napkin ring created by drilling a hole with radius \(r\) through the center of a sphere of radius \(R\) and express the answer in terms of \(h\).

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49 Applications of Integration · Level 2
Find the volume of the solid obtained by rotating about the \(y\)-axis the region bounded by \(y = 2x^2 - x^3\) and \(y = 0\).
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50 Applications of Integration · Level 2
Find the volume of the solid obtained by rotating about the \(y\)-axis the region between \(y = x\) and \(y = x^2\).
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51 Applications of Integration · Level 3
Use cylindrical shells to find the volume of the solid obtained by rotating about the \(x\)-axis the region under the curve \(y = \sqrt{x}\) from \(0\) to \(1\).
52 Applications of Integration · Level 3
Find the volume of the solid obtained by rotating the region bounded by \(y = x - x^2\) and \(y = 0\) about the line \(x = 2\).

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