Stewart 8th §6.3: Volumes by Cylindrical Shells

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Stewart 8th §6.3: Volumes by Cylindrical Shells 0/52
1 Volumes - Cylindrical Shells · 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 Volumes - Cylindrical Shells · 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 Volumes - Cylindrical Shells · 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. \(y = \sqrt[3]{x}\), \(y = 0\), \(x = 1\)
4 Volumes - Cylindrical Shells · Level 2
\(y = x^3\), \(y = 0\), \(x = 1\), \(x = 2\)
5 Volumes - Cylindrical Shells · Level 2
\(y = e^{-x^2}\), \(y = 0\), \(x = 0\), \(x = 1\)
6 Volumes - Cylindrical Shells · Level 2
\(y = 4x - x^2\), \(y = x\)
7 Volumes - Cylindrical Shells · Level 2
\(y = x^2\), \(y = 6x - 2x^2\)
8 Volumes - Cylindrical Shells · 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 Volumes - Cylindrical Shells · 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. \(x y = 1\), \(x = 0\), \(y = 1\), \(y = 3\)
10 Volumes - Cylindrical Shells · Level 3
\(y = \sqrt{x}\), \(x = 0\), \(y = 2\)
11 Volumes - Cylindrical Shells · Level 3
\(y = x^{\dfrac{3}{2}}\), \(y = 8\), \(x = 0\)
12 Volumes - Cylindrical Shells · Level 3
\(x = -3y^2 + 12y - 9\), \(x = 0\)
13 Volumes - Cylindrical Shells · Level 3
\(x = 1 + (y - 2)^2\), \(x = 2\)
14 Volumes - Cylindrical Shells · Level 3
\(x + y = 4\), \(x = y^2 - 4y + 4\)
15 Volumes - Cylindrical Shells · 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 Volumes - Cylindrical Shells · Level 3
\(y = 4 - 2x\), \(y = 0\), \(x = 0\); about \(x = -1\)
17 Volumes - Cylindrical Shells · Level 3
\(y = 4x - x^2\), \(y = 3\); about \(x = 1\)
18 Volumes - Cylindrical Shells · Level 3
\(y = \sqrt{x}\), \(x = 2y\); about \(x = 5\)
19 Volumes - Cylindrical Shells · Level 3
\(x = 2y^2\), \(y \geq 0\), \(x = 2\); about \(y = 2\)
20 Volumes - Cylindrical Shells · Level 3
\(x = 2y^2\), \(x = y^2 + 1\); about \(y = -2\)
21 Volumes - Cylindrical Shells · 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

Enter your answer directly below each part above.

22 Volumes - Cylindrical Shells · Level 3
\(y = \tan x\), \(y = 0\), \(x = \dfrac{\pi}{4}\); about \(x = \dfrac{\pi}{2}\)
23 Volumes - Cylindrical Shells · Level 3
\(y = \cos^4 x\), \(y = -\cos^4 x\), \(-\dfrac{\pi}{2} \leq x \leq \dfrac{\pi}{2}\); about \(x = \pi\)
24 Volumes - Cylindrical Shells · Level 3
\(y = x\), \(y = \dfrac{2x}{1 + x^3}\); about \(x = -1\)
25 Volumes - Cylindrical Shells · Level 3
\(x = \sqrt{\sin y}\), \(0 \leq y \leq \pi\), \(x = 0\); about \(y = 4\)
26 Volumes - Cylindrical Shells · Level 3
\(x^2 - y^2 = 7\), \(x = 4\); about \(y = 5\)
27 Volumes - Cylindrical Shells · 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 Volumes - Cylindrical Shells · 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 Volumes - Cylindrical Shells · Level 3
Each integral represents the volume of a solid. Describe the solid. \(\displaystyle\int_{0}^{3} 2 \pi x^5 d x\)
30 Volumes - Cylindrical Shells · Level 3
\(\displaystyle\int_{1}^{2} 2 \pi y \ln y d y\)
31 Volumes - Cylindrical Shells · Level 3
\(2 \pi \displaystyle\int_{1}^{4} \dfrac{y + 2}{y^2} d y\)
32 Volumes - Cylindrical Shells · Level 3
\(\displaystyle\int_{0}^{1} 2 \pi (2 - x)(3^x - 2^x) d x\)
33 Volumes - Cylindrical Shells · 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. \(y = x^2 - 2x\), \(y = \dfrac{x}{x^2 + 1}\)
34 Volumes - Cylindrical Shells · Level 4
\(y = e^{\sin x}\), \(y = x^2 - 4x + 5\)
35 Volumes - Cylindrical Shells · 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 Volumes - Cylindrical Shells · Level 4
\(y = x^3 \sin x\), \(y = 0\), \(0 \leq x \leq \pi\); about \(x = -1\)
37 Volumes - Cylindrical Shells · 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 Volumes - Cylindrical Shells · Level 3
\(y = -x^2 + 6x - 8\), \(y = 0\); about the \(x\)-axis
39 Volumes - Cylindrical Shells · Level 3
\(y^2 - x^2 = 1\), \(y = 2\); about the \(x\)-axis
40 Volumes - Cylindrical Shells · Level 3
\(y^2 - x^2 = 1\), \(y = 2\); about the \(y\)-axis
41 Volumes - Cylindrical Shells · Level 3
\(x^2 + (y - 1)^2 = 1\); about the \(y\)-axis
42 Volumes - Cylindrical Shells · Level 3
\(x = (y - 3)^2\), \(x = 4\); about \(y = 1\)
43 Volumes - Cylindrical Shells · Level 3
\(x = (y - 1)^2\), \(x - y = 1\); about \(x = -1\)
44 Volumes - Cylindrical Shells · 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 Volumes - Cylindrical Shells · Level 3
Use cylindrical shells to find the volume of the solid. A sphere of radius \(r\).
46 Volumes - Cylindrical Shells · Level 3
The solid torus of Exercise 6.2.63.
47 Volumes - Cylindrical Shells · Level 3
A right circular cone with height \(h\) and base radius \(r\).
48 Volumes - Cylindrical Shells · 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.
(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 \(h\).

Enter your answer directly below each part above.

49 Volumes - Cylindrical Shells · Level 3
Find the volume of the solid obtained by rotating about the \(y\)-axis the region bounded by \(y = 2x^2 - x^3\) and \(y = 0\).
50 Volumes - Cylindrical Shells · Level 2
Find the volume of the solid obtained by rotating about the \(y\)-axis the region between \(y = x\) and \(y = x^2\).
51 Volumes - Cylindrical Shells · 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 Volumes - Cylindrical Shells · 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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