2.2E: Exercises for Section 6.2
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- May 24, 2023
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- Gilbert Strang & Edwin “Jed” Herman
- OpenStax
( \newcommand{\kernel}{\mathrm{null}\,}\)
1) Derive the formula for the volume of a sphere using the slicing method.
2) Use the slicing method to derive the formula for the volume of a cone.
3) Use the slicing method to derive the formula for the volume of a tetrahedron with side length a.
4) Use the disk method to derive the formula for the volume of a trapezoidal cylinder.
5) Explain when you would use the disk method versus the washer method. When are they interchangeable?
Volumes by Slicing
For exercises 6 - 10, draw a typical slice and find the volume using the slicing method for the given volume.
6) A pyramid with height 6 units and square base of side 2 units, as pictured here.
- Solution:
- Here the cross-sections are squares taken perpendicular to the y-axis.
We use the vertical cross-section of the pyramid through its center to obtain an equation relating x and y.
Here this would be the equation, y=6−6x. Since we need the dimensions of the square at each y-level, we solve this equation for x to get, x=1−y6.
This is half the distance across the square cross-section at the y-level, so the side length of the square cross-section is, s=2(1−y6).
Thus, we have the area of a cross-section is,
A(y)=[2(1−y6)]2=4(1−y6)2.
Then,V=∫604(1−y6)2dy=−24∫01u2du,whereu=1−y6,sodu=−16dy,⟹−6du=dy=24∫10u2du=24u33|10=8u3|10=8(13−03)=8units3
7) A pyramid with height 4 units and a rectangular base with length 2 units and width 3 units, as pictured here.
8) A tetrahedron with a base side of 4 units,as seen here.
- Answer
- V=323√2=16√23 units3
9) A pyramid with height 5 units, and an isosceles triangular base with lengths of 6 units and 8 units, as seen here.
10) A cone of radius r and height h has a smaller cone of radius r/2 and height h/2 removed from the top, as seen here. The resulting solid is called a frustum.
- Answer
- V=7π12hr2 units3
For exercises 11 - 16, draw an outline of the solid and find the volume using the slicing method.
11) The base is a circle of radius a. The slices perpendicular to the base are squares.
12) The base is a triangle with vertices (0,0),(1,0), and (0,1). Slices perpendicular to the xy-plane are semicircles.
- Answer
-
V=∫10π(1−x)28dx=π24 units3
13) The base is the region under the parabola y=1−x2 in the first quadrant. Slices perpendicular to the xy-plane are squares.
14) The base is the region under the parabola y=1−x2 and above the x-axis. Slices perpendicular to the y-axis are squares.
- Answer
-
V=∫104(1−y)dy=2 units3
15) The base is the region enclosed by y=x2 and y=9. Slices perpendicular to the x-axis are right isosceles triangles.
16) The base is the area between y=x and y=x2. Slices perpendicular to the x-axis are semicircles.
- Answer
-
V=∫10π8(x−x2)2dx=π240 units3
Disk and Washer Method
For exercises 17 - 24, draw the region bounded by the curves. Then, use the disk or washer method to find the volume when the region is rotated around the x-axis.
17) x+y=8,x=0, and y=0
18) y=2x2,x=0,x=4, and y=0
- Answer
-
V=∫404πx4dx=4096π5 units3
19) y=ex+1,x=0,x=1, and y=0
20) y=x4,x=0, and y=1
- Answer
-
V=∫10π(12−(x4)2)dx=∫10π(1−x8)dx=8π9 units3
21) y=√x,x=0,x=4, and y=0
22) y=sinx,y=cosx, and x=0
- Answer
-
V=∫π/40π(cos2x−sin2x)dx=∫π/40πcos2xdx=π2 units3
23) y=1x,x=2, and y=3
24) x2−y2=9 and x+y=9,y=0 and x=0
- Answer
-
V=207π units3
For exercises 25 - 32, draw the region bounded by the curves. Then, find the volume when the region is rotated around the y-axis.
25) y=4−12x,x=0, and y=0
26) y=2x3,x=0,x=1, and y=0
- Answer
-
V=4π5 units3
27) y=3x2,x=0, and y=3
28) y=√4−x2,y=0, and x=0
- Answer
-
V=16π3 units3
29) y=1√x+1,x=0, and x=3
30) x=sec(y) and y=π4,y=0 and x=0
- Answer
-
V=π units3
31) y=1x+1,x=0, and x=2
32) y=4−x,y=x, and x=0
- Answer
-
V=16π3 units3
For exercises 33 - 40, draw the region bounded by the curves. Then, find the volume when the region is rotated around the x-axis.
33) y=x+2,y=x+6,x=0, and x=5
34) y=x2 and y=x+2
- Answer
-
V=72π5 units3
35) x2=y3 and x3=y2
36) y=4−x2 and y=2−x
- Answer
-
V=108π5 units3
37) [T] y=cosx,y=e−x,x=0, and x=1.2927
38) y=√x and y=x2
- Answer
-
V=3π10 units3
39) y=sinx,y=5sinx,x=0 and x=π
40) y=√1+x2 and y=√4−x2
- Answer
-
V=2√6π units3
For exercises 41 - 45, draw the region bounded by the curves. Then, use the washer method to find the volume when the region is revolved around the y-axis.
41) y=√x,x=4, and y=0
42) y=x+2,y=2x−1, and x=0
- Answer
-
V=9π units3
43) y=3x and y=x3
44) x=e2y,x=y2,y=0, and y=ln(2)
- Answer
-
V=π20(75−4ln5(2)) units3
45) x=√9−y2,x=e−y,y=0, and y=3
46) Yogurt containers can be shaped like frustums. Rotate the line y=(1m)x around the y-axis to find the volume between y=a and y=b.
- Answer
- V=m2π3(b3−a3) units3
47) Rotate the ellipse x2a2+y2b2=1 around the x-axis to approximate the volume of a football, as seen here.
48) Rotate the ellipse x2a2+y2b2=1 around the y-axis to approximate the volume of a football.
- Answer
- V=4a2bπ3 units3
49) A better approximation of the volume of a football is given by the solid that comes from rotating y=sinx around the x-axis from x=0 to x=π. What is the volume of this football approximation, as seen here?
For exercises 51 - 56, find the volume of the solid described.
51) The base is the region between y=x and y=x2. Slices perpendicular to the x-axis are semicircles.
52) The base is the region enclosed by the generic ellipse x2a2+y2b2=1. Slices perpendicular to the x-axis are semicircles.
- Answer
- V=2ab2π3 units3
53) Bore a hole of radius a down the axis of a right cone and through the base of radius b, as seen here.
54) Find the volume common to two spheres of radius r with centers that are 2h apart, as shown here.
- Answer
- V=π12(r+h)2(6r−h) units3
55) Find the volume of a spherical cap of height h and radius r where h<r, as seen here.
56) Find the volume of a sphere of radius R with a cap of height h removed from the top, as seen here.
- Answer
- V=π3(h+R)(h−2R)2 units3
Contributors
Gilbert Strang (MIT) and Edwin “Jed” Herman (Harvey Mudd) with many contributing authors. This content by OpenStax is licensed with a CC-BY-SA-NC 4.0 license. Download for free at http://cnx.org.