Calculus2
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Volume.pdf
Volume
� 1–3 For the solid S described, do the following:
a) Sketch the base of S in the xy-plane.
b) Sketch a three-dimensional picture of S with the xy-plane as the floor.
c) Compute the volume of S.
1. The base of S is the region lying above the parabola y = x2 and below the line y = 1 over the interval 0 ≤ x ≤ 1. Cross-sections perpendicular to the x-axis are squares.
2. The base of S is the same region as in Problem (1). Cross-sections perpendicular to the y-axis are equilateral triangles.
Hint: Integrate the cross-sectional area A(y) with respect to y. The area of an equilateral triangle of
side length a is √ 3 4 a2.
3. The base of S is the triangular region with vertices (0, 0), (3, 0), and (0, 2). Cross-sections perpendicular to the y-axis are semicircles.
Solids of Revolution
� 4–7 Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Include a sketch of the solid and a typical disk or washer.
4. y = 1/x, x = 1, x = 2, y = 0; about the x-axis
5. y = x3, y = 8, x = 0; about the y-axis
6. y = sin x, y = 1, x = 0; about the x-axis
7. y = ln x, x = e, y = 0; about the y-axis
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8. Prove that the volume of a cone of height h and base radius r is given by V = 1
3 πr2h.
Hint: Position the cone as shown:
x
y
h
r
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� 9–12 Set up, BUT DON’T COMPUTE, the volume integral for the solid obtained by revolving the region described about the specified line.
9. y = cos x over 0 ≤ x ≤ π/2; about the line x = −1
10. The same region as in Problem (9); about the line y = 2
11. y = ex, x = 0, x = 1, y = 0; about the line y = −1
12. The same region as in Problem (11); about the line x = 2
The Method of Shells
� 13, 14 Use the Method of Shells to find the volume of the solid described in the indicated problem.
13. Problem (4)
14. Problem (7)
� 15, 16 Use the Method of Shells to set up, BUT NOT COMPUTE, the volume integral for the solid described in the indicated problem.
15. Problem (10)
16. Problem (12)
� Challenge Problem1
Use the Method of Shells to prove that the volume of a torus (donut) with major radius R and minor radius r is given by V = 2π2r2R.
Hint: Center the torus at the origin.
r
R
1Challenge problems are not graded—they are offered for your personal edification. I’d love to see your solution!
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Solutions to Selected Problems
1. 8 15
2. √ 3 8
3. 3π 4
4. π 2
5. 96 5 π
6. π 2
4
7. π 2
(e2 + 1)
8. Proof
9. π ∫ 1 0
[ (cos−1 y + 1)2 − 12
] dy
10. π ∫ π
2
0
[ 4 − (2 − cos x)2
] dx
11. π ∫ 1 0
[ (ex + 1)2 − 12
] dx
12. π ∫ 1 0
( 22 − 12
) dy + π
∫ e 1
[ (2 − ln y)2 − 12
] dy
13. π 2
14. π 2
(e2 + 1)
15. ∫ 1 0
2π(2 − y) cos−1 y dy
16. ∫ 1 0
2π(2 − x)ex dx
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