Stewart 8th §6.2: Volumes

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Stewart 8th §6.2: Volumes 0/81
1 Applications of Integration · Level 2
Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer. \(y = x + 1\), \(y = 0\), \(x = 0\), \(x = 2\); about the \(x\)-axis
2 Applications of Integration · Level 2
\(y = \dfrac{1}{x}\), \(y = 0\), \(x = 1\), \(x = 4\); about the \(x\)-axis
3 Applications of Integration · Level 2
\(y = \sqrt{x - 1}\), \(y = 0\), \(x = 5\); about the \(x\)-axis
4 Applications of Integration · Level 2
\(y = e^x\), \(y = 0\), \(x = -1\), \(x = 1\); about the \(x\)-axis
5 Applications of Integration · Level 2
\(x = 2 \sqrt{y}\), \(x = 0\), \(y = 9\); about the \(y\)-axis
6 Applications of Integration · Level 2
\(2x = y^2\), \(x = 0\), \(y = 4\); about the \(y\)-axis
7 Applications of Integration · Level 3
\(y = x^3\), \(y = x\), \(x \geq 0\); about the \(x\)-axis
8 Applications of Integration · Level 3
\(y = 6 - x^2\), \(y = 2\); about the \(x\)-axis
9 Applications of Integration · Level 3
\(y^2 = x\), \(x = 2y\); about the \(y\)-axis
10 Applications of Integration · Level 3
\(x = 2 - y^2\), \(x = y^4\); about the \(y\)-axis
11 Applications of Integration · Level 3
\(y = x^2\), \(x = y^2\); about \(y = 1\)
12 Applications of Integration · Level 3
\(y = x^3\), \(y = 1\), \(x = 2\); about \(y = -3\)
13 Applications of Integration · Level 3
\(y = 1 + \sec x\), \(y = 3\); about \(y = 1\)
14 Applications of Integration · Level 3
\(y = \sin x\), \(y = \cos x\), \(0 \leq x \leq \dfrac{\pi}{4}\); about \(y = -1\)
15 Applications of Integration · Level 3
\(y = x^3\), \(y = 0\), \(x = 1\); about \(x = 2\)
16 Applications of Integration · Level 3
\(x y = 1\), \(y = 0\), \(x = 1\), \(x = 2\); about \(x = -1\)
17 Applications of Integration · Level 3
\(x = y^2\), \(x = 1 - y^2\); about \(x = 3\)
18 Applications of Integration · Level 3
\(y = x\), \(y = 0\), \(x = 2\), \(x = 4\); about \(x = 1\)
19 Applications of Integration · Level 3
Refer to the figure and find the volume generated by rotating the given region about the specified line. [Figure: \(y = \sqrt[4]{x}\) in \(cal(R)_1\), \(cal(R)_2\), \(cal(R)_3\) with corners \(O(0,0)\), \(A(1,0)\), \(B(1,1)\), \(C(0,1)\)] \(cal(R)_1\) about \(O A\)
20 Applications of Integration · Level 3
\(cal(R)_1\) about \(O C\)
21 Applications of Integration · Level 3
\(cal(R)_1\) about \(A B\)
22 Applications of Integration · Level 3
\(cal(R)_1\) about \(B C\)
23 Applications of Integration · Level 3
\(cal(R)_2\) about \(O A\)
24 Applications of Integration · Level 3
\(cal(R)_2\) about \(O C\)
25 Applications of Integration · Level 3
\(cal(R)_2\) about \(A B\)
26 Applications of Integration · Level 3
\(cal(R)_2\) about \(B C\)
27 Applications of Integration · Level 3
\(cal(R)_3\) about \(O A\)
28 Applications of Integration · Level 3
\(cal(R)_3\) about \(O C\)
29 Applications of Integration · Level 3
\(cal(R)_3\) about \(A B\)
30 Applications of Integration · Level 3
\(cal(R)_3\) about \(B C\)
31 Applications of Integration · Level 3
Set up an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Then use your calculator to evaluate the integral correct to five decimal places. \(y = e^{-x^2}\), \(y = 0\), \(x = -1\), \(x = 1\)
(a) About the \(x\)-axis
(b) About \(y = -1\)

Enter your answer directly below each part above.

32 Applications of Integration · Level 3
\(y = 0\), \(y = \cos^2 x\), \(-\dfrac{\pi}{2} \leq x \leq \dfrac{\pi}{2}\)
(a) About the \(x\)-axis
(b) About \(y = 1\)

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33 Applications of Integration · Level 3
\(x^2 + 4y^2 = 4\)
(a) About \(y = 2\)
(b) About \(x = 2\)

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34 Applications of Integration · Level 3
\(y = x^2\), \(x^2 + y^2 = 1\), \(y \geq 0\)
(a) About the \(x\)-axis
(b) About the \(y\)-axis

Enter your answer directly below each part above.

35 Applications of Integration · Level 4
Use a graph to find approximate \(x\)-coordinates of the points of intersection of the given curves. Then use your calculator to find (approximately) the volume of the solid obtained by rotating about the \(x\)-axis the region bounded by these curves. \(y = \ln(x^6 + 2)\), \(y = \sqrt{3 - x^3}\)
36 Applications of Integration · Level 4
\(y = 1 + x e^{-x^3}\), \(y = \arctan x^2\)
37 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. \(y = \sin^2 x\), \(y = 0\), \(0 \leq x \leq \pi\); about \(y = -1\)
38 Applications of Integration · Level 4
\(y = x\), \(y = x e^{1 - \dfrac{x}{2}}\); about \(y = 3\)
39 Applications of Integration · Level 3
Each integral represents the volume of a solid. Describe the solid. \(\pi \displaystyle\int_{0}^{\pi} \sin x d x\)
40 Applications of Integration · Level 3
Each integral represents the volume of a solid. Describe the solid. \(\pi \displaystyle\int_{-1}^1 (1 - y^2)^2 d y\)
41 Applications of Integration · Level 3
Each integral represents the volume of a solid. Describe the solid. \(\pi \displaystyle\int_{0}^{1} (y^4 - y^8) d y\)
42 Applications of Integration · Level 3
Each integral represents the volume of a solid. Describe the solid. \(\pi \displaystyle\int_{1}^{4} [3^2 - (3 - \sqrt{x})^2] d x\)
43 Applications of Integration · Level 3
A CAT scan produces equally spaced cross-sectional views of a human organ that provide information about the organ otherwise obtained only by surgery. Suppose that a CAT scan of a human liver shows cross-sections spaced 1.5 cm apart. The liver is 15 cm long and the cross-sectional areas, in square centimeters, are 0, 18, 58, 79, 94, 106, 117, 128, 63, 39, and 0. Use the Midpoint Rule to estimate the volume of the liver.
44 Applications of Integration · Level 3
A log 10 m long is cut at 1-meter intervals and its cross-sectional areas \(A\) (at a distance \(x\) from the end of the log) are listed in the table. Use the Midpoint Rule with \(n = 5\) to estimate the volume of the log.
\(x\) (m) \(A\) (m\(^2\)) \(x\) (m) \(A\) (m\(^2\))
0 0.68 6 0.53
1 0.65 7 0.55
2 0.64 8 0.52
3 0.61 9 0.50
4 0.58 10 0.48
5 0.59
45 Applications of Integration · Level 3
(a) If the region shown in the figure is rotated about the \(x\)-axis to form a solid, use the Midpoint Rule with \(n = 4\) to estimate the volume of the solid. [Figure with curve from \(x = 2\) to \(x = 10\), height up to \(y = 4\)]
(b) Estimate the volume if the region is rotated about the \(y\)-axis. Again use the Midpoint Rule with \(n = 4\).

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46 Applications of Integration · Level 4
(a) A model for the shape of a bird's egg is obtained by rotating about the \(x\)-axis the region under the graph of \(f(x) = (a x^3 + b x^2 + c x + d) \sqrt{1 - x^2}\). Use a CAS to find the volume of such an egg.
(b) For a red-throated loon, \(a = -0.06\), \(b = 0.04\), \(c = 0.1\), and \(d = 0.54\). Graph \(f\) and find the volume of an egg of this species.

Enter your answer directly below each part above.

47 Applications of Integration · Level 3
Find the volume of the described solid \(S\). A right circular cone with height \(h\) and base radius \(r\).
48 Applications of Integration · Level 3
Find the volume of the described solid \(S\). A frustum of a right circular cone with height \(h\), lower base radius \(R\), and top radius \(r\).
49 Applications of Integration · Level 4
Find the volume of the described solid \(S\). A cap of a sphere with radius \(r\) and height \(h\).
50 Applications of Integration · Level 4
Find the volume of the described solid \(S\). A frustum of a pyramid with square base of side \(b\), square top of side \(a\), and height \(h\). What happens if \(a = b\)? What happens if \(a = 0\)?
51 Applications of Integration · Level 3
A pyramid with height \(h\) and rectangular base with dimensions \(b\) and \(2b\).
52 Applications of Integration · Level 3
A pyramid with height \(h\) and base an equilateral triangle with side \(a\) (a tetrahedron).
53 Applications of Integration · Level 3
A tetrahedron with three mutually perpendicular faces and three mutually perpendicular edges with lengths 3 cm, 4 cm, and 5 cm.
54 Applications of Integration · Level 3
The base of \(S\) is a circular disk with radius \(r\). Parallel cross-sections perpendicular to the base are squares.
55 Applications of Integration · Level 3
The base of \(S\) is an elliptical region with boundary curve \(9x^2 + 4y^2 = 36\). Cross-sections perpendicular to the \(x\)-axis are isosceles right triangles with hypotenuse in the base.
56 Applications of Integration · Level 3
The base of \(S\) is the triangular region with vertices \((0, 0)\), \((1, 0)\), and \((0, 1)\). Cross-sections perpendicular to the \(y\)-axis are equilateral triangles.
57 Applications of Integration · Level 3
The base of \(S\) is the same base as in Exercise 56, but cross-sections perpendicular to the \(x\)-axis are squares.
58 Applications of Integration · Level 3
The base of \(S\) is the region enclosed by the parabola \(y = 1 - x^2\) and the \(x\)-axis. Cross-sections perpendicular to the \(y\)-axis are squares.
59 Applications of Integration · Level 3
The base of \(S\) is the same base as in Exercise 58, but cross-sections perpendicular to the \(x\)-axis are isosceles triangles with height equal to the base.
60 Applications of Integration · Level 3
The base of \(S\) is the region enclosed by \(y = 2 - x^2\) and the \(x\)-axis. Cross-sections perpendicular to the \(y\)-axis are quarter-circles.
61 Applications of Integration · Level 4
The solid \(S\) is bounded by circles that are perpendicular to the \(x\)-axis, intersect the \(x\)-axis, and have centers on the parabola \(y = \dfrac{1}{2}(1 - x^2)\), \(-1 \leq x \leq 1\).
62 Applications of Integration · Level 4
The base of \(S\) is a circular disk with radius \(r\). Parallel cross-sections perpendicular to the base are isosceles triangles with height \(h\) and unequal side in the base.
(a) Set up an integral for the volume of \(S\).
(b) By interpreting the integral as an area, find the volume of \(S\).

Enter your answer directly below each part above.

63 Applications of Integration · Level 4
(a) Set up an integral for the volume of a solid torus (the donut-shaped solid shown in the figure) with radii \(r\) and \(R\).
(b) By interpreting the integral as an area, find the volume of the torus.

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64 Applications of Integration · Level 4
Solve Example 9 taking cross-sections to be parallel to the line of intersection of the two planes.
65 Applications of Integration · Level 4
(a) Cavalieri's Principle states that if a family of parallel planes gives equal cross-sectional areas for two solids \(S_1\) and \(S_2\), then the volumes of \(S_1\) and \(S_2\) are equal. Prove this principle.
(b) Use Cavalieri's Principle to find the volume of the oblique cylinder shown in the figure.

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66 Applications of Integration · Level 5
Find the volume common to two circular cylinders, each with radius \(r\), if the axes of the cylinders intersect at right angles.
67 Applications of Integration · Level 5
Find the volume common to two spheres, each with radius \(r\), if the center of each sphere lies on the surface of the other sphere.
68 Applications of Integration · Level 4
A bowl is shaped like a hemisphere with diameter 30 cm. A heavy ball with diameter 10 cm is placed in the bowl and water is poured into the bowl to a depth of \(h\) centimeters. Find the volume of water in the bowl.
69 Applications of Integration · Level 4
A hole of radius \(r\) is bored through the middle of a cylinder of radius \(R > r\) at right angles to the axis of the cylinder. Set up, but do not evaluate, an integral for the volume cut out.
70 Applications of Integration · Level 4
A hole of radius \(r\) is bored through the center of a sphere of radius \(R > r\). Find the volume of the remaining portion of the sphere.
71 Applications of Integration · Level 5
Some of the pioneers of calculus, such as Kepler and Newton, were inspired by the problem of finding the volumes of wine barrels. (In fact Kepler published a book *Stereometria doliorum* in 1615 devoted to methods for finding the volumes of barrels.) They often approximated the shape of the sides by parabolas.
(a) A barrel with height \(h\) and maximum radius \(R\) is constructed by rotating about the \(x\)-axis the parabola \(y = R - c x^2\), \(-\dfrac{h}{2} \leq x \leq \dfrac{h}{2}\), where \(c\) is a positive constant. Show that the radius of each end of the barrel is \(r = R - d\), where \(d = \dfrac{c h^2}{4}\).
(b) Show that the volume enclosed by the barrel is \(V = \dfrac{1}{3} \pi h \left(2 R^2 + r^2 - \dfrac{2}{5} d^2\right)\).

Enter your answer directly below each part above.

72 Applications of Integration · Level 5
Suppose that a region \(cal(R)\) has area \(A\) and lies above the \(x\)-axis. When \(cal(R)\) is rotated about the \(x\)-axis, it sweeps out a solid with volume \(V_1\). When \(cal(R)\) is rotated about the line \(y = -k\) (where \(k\) is a positive number), it sweeps out a solid with volume \(V_2\). Express \(V_2\) in terms of \(V_1\), \(k\), and \(A\).
73 Applications of Integration · Level 3
Show that the volume of a sphere of radius \(r\) is \(V = \dfrac{4}{3} \pi r^3\).
74 Applications of Integration · Level 2
Find the volume of the solid obtained by rotating about the \(x\)-axis the region under the curve \(y = \sqrt{x}\) from \(0\) to \(1\). Illustrate the definition of volume by sketching a typical approximating cylinder.
75 Applications of Integration · Level 2
Find the volume of the solid obtained by rotating the region bounded by \(y = x^3\), \(y = 8\), and \(x = 0\) about the \(y\)-axis.
76 Applications of Integration · Level 3
The region \(cal(R)\) enclosed by the curves \(y = x\) and \(y = x^2\) is rotated about the \(x\)-axis. Find the volume of the resulting solid.
77 Applications of Integration · Level 3
Find the volume of the solid obtained by rotating the region in Example 4 about the line \(y = 2\).
78 Applications of Integration · Level 3
Find the volume of the solid obtained by rotating the region in Example 4 about the line \(x = -1\).
79 Applications of Integration · Level 3
A solid has a circular base of radius 1. Parallel cross-sections perpendicular to the base are equilateral triangles. Find the volume of the solid.
80 Applications of Integration · Level 3
Find the volume of a pyramid whose base is a square with side \(L\) and whose height is \(h\).
81 Applications of Integration · Level 4
A wedge is cut out of a circular cylinder of radius 4 by two planes. One plane is perpendicular to the axis of the cylinder. The other intersects the first at an angle of \(30^{\circ}\) along a diameter of the cylinder. Find the volume of the wedge.

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