Contents
12 Applications of the Definite Integral
179
12.1 Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
12.2 Solids of Revolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
12.3 Surface Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
12.4 Arc Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
12.5 Liquid Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
12.6 Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
2
CONTENTS
Chapter 12
Applications of the Definite Integral
12.1
Area
In problems 1 to 15 find the area of the region under the following curves between the given values of x. Use
the principles of graph sketching taught in your course to sketch the required region. Label your diagram.
1. y = 5 − 12 x; x = −6, x = 0
2. y = 12 x2 ; x = 0, x = 3
3. y = x2 + 3; x = −2, x = 2
4. y = 10x − x2 ; x = 1, x = 10
5. y =
√
x; x = 0, x = 4
7. y = x(x − 5)2 ; x = 0, x = 2
9. y = 2x +
1
x2 ;
x = 1, x = 3
√
11. y = x 4x2 + 1; x = 0, x = 2
13. y =
(
14. y =
(
(x + 1)3 , −1 ≤ x ≤ 0,
1−
x2
4 ,
0 < x ≤ 2;
1
2 (x
+ 3), −3 ≤ x < 1,
3 − x,
1 ≤ x ≤ 3;
√
6. y = 2 x − 1; x = 1, x = 10
8. y = (x + 1)3 + 1; x = −2, x = 0
10. y =
√5 ;
x+2
12. y =
x
c
√
x = 0, x = 5
c2 − x2 ; x = c, where c > 0
x = −1, x = 2
x = −3, x = 3
15. y = |x|; x = −2, x = 5
In problems 16 to 64 find the area of the region or regions bounded by the following curves. Use the principles
of graph sketching taught in your course to sketch the required region(s). Label your diagram.
16. y = x3 , y = 0, x = 1, x = 2
17. y = x2 + 4x, y = 0
18. y = x2 − 4x + 2, x + y − 6 = 0
19. y = −x2 − 6x, y = 2x
20. y = x2 − 6, y = 6 − z 2
21. y = x3 , y = 1, x = 0, x = 3
180
Applications of the Definite Integral
22. y = x(x − 2)2 , y = 0
√
24. y = 4 + x, x = 0, y = 0
23. y = x(x + 3)2 , y = 0, x = −3, x = −1
25. y = x3 − x2 − 6x, y = 0
26. x3 − 2x2 − 11x + 12, y = 0
√
28. y = 2 x, y = −2x + 12, y = 0
27. y = 9 − x2 , x − y + 7 = 0
√
29. y = 2 x, y = −2x + 12, x = 0
30. y = −x3 + 2x2 + 3x, y = −5x
31. x2 y = 3, 4x + 3y − 13 = 0
32. y 2 − 4y − 2x = 0, x = 0
33. y 2 = x + 1, y 2 = 7 − x
34. y 2 = x + 4, x − 2y + 1 = 0
35. x = 4y − y 2 , x + y = 4
36. x = y(y − 3)2 , x = 0
37. x = y 3 − 4y 2 − y + 4, x = 0
38. x = y 2 − y 3 , x = 0
39. y 2 = x3 , x = 0, x = 1
40. y 2 = ax, x2 = by, where a > 0 and b > 0
41. y = x4 − 3x2 , y = x2
√
√
√
43.
x + y = a, x = 0, y = 0
42. y 2 = x2 (x2 − 1), x = 3
√
44. y = x, y = x3
45. x = 1, x = 2, y = x2 , x2 y = 1
46. y = |x|, y = x3
47. y = tan x, y = 0, x = π4
√
49. y = ex , y = x + 1, x = 1
48. y = ex , x = 0, x = 1
51. y = e−x , y = x + 1, x = −1
50. y = ex , y = 10x , y = e
√
52. y = e−x , y = x + 1, x = −1
53. y = e−x , y = e
54. y = 12 (ex + e−x ), y = 2
55. y = 12 (ex − e−x ), y = x, x = −1
56. y = xe−x , y = 0, x = 0, and x = c; where c is the x coordinate of the curve’s inflection point.
57. y = (1 − x)e−x , y = 0, x = 0
58. y = (x − 2)e−x , y = 0, x = 0
59. y = x2 e−x , y = 0, x = c; where c is the x coordinate of the relative maximum point of y.
60. y = x ln x, y = 0
61. y = e|x| , y = e
62. y = ln x, y = 0, x = e
63. y = sin x, y = cos x, x = 0, and x = 10, 000π
64.
y = sin x and the tangent lines at the inflection points that occur in the interval [0, 2π].
65.
Find the area under one arch of the curve y = a sin(x/a).
66.
Find the area bounded by the curve y = xe−x , the x-axis, and the line x = c, where y(c) is maximum.
1
[(1 − e− 2 )/2]
67.
Find the area bounded by the curve y = (ln x)/x, the x-axis, and the line x = c, where y(c) is maximum.
68.
Find the area bounded by the curve y = (ln x − 1)/x, the x-axis and the line x = c, where y(c) is
maximum.
69.
Show that the area bounded by the curve y = (ln x − c)/x, the x-axis and the vertical line through the
maximum point of the curve is independent of the constant c.
70.
Find the area under the first arch of the curve y = x sin x.
2
12.2 Solids of Revolution
12.2
181
Solids of Revolution
Every solution must include a neat labelled diagram. Use your diagram to set up the appropriate definite
integral instead of picking a formula from your text book and hoping for the best.
71.
The area bounded by y = e−x the axes, and the line x = 2 is revolved about the x-axis. Find the
volume generated.
72.
The area under one arch of the sine curve revolved about the x-axis. Find the volume generated.
73.
Find the volume formed by revolving the area defined in question 71 about the line y = 1.
74.
The area bounded by the parabola y = x2 /a the x-axis, and the line x = b is revolved about the x-axis.
h 5i
πb
Find the volume generated.
5a2
75.
The area defined in question 74 is revolved about the y-axis. Find the volume generated.
76.
The area defined in question 74 is revolved about the line x = b. Find the volume generated.
77.
Find the volume generated by rotating an ellipse about its major axis.
78.
Find the volume generated by rotating an ellipse about its minor axis.
In problems 79 to 90 find the volumes of the solids, V , generated by revolving the described regions, R,
about the x axis. A labelled diagram is required as usual.
79.
80.
R is bounded by x + y = 3, the coordinate axis.
√
R is bounded by y = 2 5x, x = 4 and the x-axis.
81.
R is bounded by the axes and the line 2x + 3y = 1.
82.
R is bounded by axis and the line (x/h) + (y/r) = 1, h > 0 and r > 0.
83.
R is bounded by y = 0, and y = 4 − x2 .
84.
R is bounded by y = 0, x = 2, x = 4 − y 2 .
85.
86.
R is bounded by x = π/3, y = 0, and y = tan x.
√
R is bounded by y = x/ 4 − x, y = 0, x = 3.
87.
R is bounded by y = 0, y = ln x, x = e.
88.
R is bounded by y = xex , y = 0, x = 1.
89.
R is bounded by the y-axis, the line y = 1, and that arc of y = sin x between x = 0 and x = π/2.
90.
R is bounded by x + y = 5, xy = 4.
In problems 91 to 102 find the volume generated by the region, R, rotated about the given axis. Use the
method of shells and include a neat diagram for each problem.
91.
R is the region in the first quadrant bounded by y = 4 − x2 , and the x and y axes. Rotated about the
y-axis.
92.
R is the region defined in problem 91 rotated about the line x = 2.
182
Applications of the Definite Integral
93.
R is the region bounded by y = sin x and the interval [0, π] on the x axis rotated about the y axis.
94.
R is the region bounded by x = 1, y = e−x , and the axes rotated about the line x = 1.
√
R is bounded by y = x, x = 0 and y = 2 rotated about the y axis.
95.
96.
97.
98.
99.
R is bounded by y = 0, y = ln x, x = e rotated about the y-axis.
p
R is bounded by the x-axis, the y-axis, the line y = 2 and x = y 2 + 4 rotated about the y-axis.
R is bounded below by y = 0 and above by the curves x = y 2 and x = 8 − y 2 rotated about the y-axis.
√
R is bounded by y = x and y = x3 rotated about the y-axis.
100. A cylindrical hole is bored through a golden sphere of radius b to form a wedding ring. If the length
of the hole is ` where `/2 < b compute the volume of the ring. What is remarkable about this result?
(No funny answers please.)
2
101. R is the region y = e−x , y = 0 and x = 0 rotated about the y axis.
102. R is the region y = ex , y = e−x , and x = 1 rotated about the y axis.
12.3
Surface Area
103. Find the area of the surface of a right circular cone of radius r and altitude h.
104. Find the areas of the surface obtained by revolving the part of the curve y =
x = 2 and x = 6 around the x axis.
√
x that lies between
105. Let a sphere be inside a circular cylinder whose radius equals the radius of the sphere. If two planes
cut the cylinder at right angles to its axis, and intersect the sphere, show that the area on the sphere
between the planes is the same as the area on the cylinder between the planes.
In problems 106 to 124 find the surface area generated by rotating the given arc of a curve about the given
line.
106. y = x3 (0 ≤ x ≤ 2/3); about y = 0.
107. y = (x3 + 3/x)/6, (1 ≤ x ≤ 3), about y = 0.
108. y = (x3 + 48/x)/24, (2 ≤ x ≤ 4), about y = 0.
√
109. y = x(4x − 3)/6, (1 ≤ x ≤ 9), about y = 0.
√
110. y = x|x − 3|/3, (1 ≤ x ≤ 4), about y = 0.
111. y = (x4 + 2/x2 )/8, (1 ≤ x ≤ 3), about y = 0.
112. y = 81 (2x4 + 1/x2 ), (1 ≤ x ≤ 2), about y = 0.
113. y = x2 , (0 ≤ x ≤ 1), about y = 0.
114. y = x2 , (0 ≤ x ≤ 1), about x = 0.
115. y = (x3 /3) + 1/4x, (1 ≤ x ≤ 2), about x = 0.
116. y = (x3 /3) + 1/4x, (1 ≤ x ≤ 2), about y = 0.
117. x2 + y 2 = a2 , about y = a.
[47π/4]
12.4 Arc Length
183
3
118. y = (x2 + 2) 2 /3, (0 ≤ x ≤ 3), about x = 0.
119. y = x3 /3 + 1/4x, (1 ≤ x ≤ 3), about y = −1.
120. y = sin x, (0 ≤ x ≤ π), about y = 0.
121. y = ex , (0 ≤ x ≤ 1), about y = 0.
122. y = ln x, (1 ≤ x ≤ 2), about x = 0.
123. y = (ex + e−x )/2, (−1 ≤ x ≤ 1), about y = 0.
124. x2 − y 2 = a2 , (0 ≤ y ≤ 2a), about x = 0.
12.4
Arc Length
Find the arc length of each curve between the points indicated in the following problems. Sketch the curve
between the indicated points for each problem.
125. (a) y = 2x + 1
from x = 1 to x = 2
(b) y = 2x + k
from x = 1 to x = 2
where k is any constant
126. (a) y = mx + k
from x = 1 to x = 2
(b) y = mx + k
from x = a to x = b
127. y = 16 (x3 + 3/x)
from x = 1 to x = 3
128. y =
129. y =
130. y =
1
3
24 (x
√
√
+ 48/x)
from x = 2 to x = 4
x(4x − 3)/6
from x = 1 to x = 9
x(x − 3)/3
from x = 1 to x = 4
131. y = (x4 + 2/x2 )/8
from x = 1 to x = 3
132. y = (4x4 + 1/x2 )/8
from x = 1 to x = 2
133. y 3 = 8x2
from (1, 2) to (8, 8)
134. y 2 = x3
from (0, 0) to (1, 1)
135. x 3 + y 3 = 4
√
√
from (2 2, 2 2) and (8, 0)
136. 4y = x2
from (−2, 1) to (4, 4)
137. y = ln x
from x =
2
2
3
1
2
to x = 2
138. y = x 2
from x = 0 to x = 4
139. y = ln(cos x)
from x = 0 to x = π/3
140. y = (ex + e−x )/2
from x = −1 to x = 1
17 6
h
i
√
(104 13−125)
27
184
Applications of the Definite Integral
141. (y + 1)2 = 4x3
from (0, −1) to (1, 1)
142. x2 + 2y + 2 = 0
√
from (− 2, −2) to (0, −1)
143. y 2 = (x2 /2) − (ln x)/4
from x = 1 to x = 2
144. y = ln(1 − x2 )
from x = 0 to x = 3/4
Find the arc length of each curve between the point indicated. Sketch the curve as usual. Integrate with
respect to y.
145. y 2 = −4x
from (−4, 4) to (0, 0)
146. 3x2 = y 3
√
from (−3, 3) to (8/ 3, 4)
147. the shorter arc of x2 + y 2 = 32
√
√
from (4, 4) to (2 6, −2 2)
148. y = e−2x
from y =
149. y = sin−1 (ex )
from y = π/6 to y = π/2
2
150. y = x 3
2
151. y = x 3
12.5
1
4
to y = 4
from x = −1 to x = 8
from x = 1 to x = 27
Liquid Pressure
In the following problems a neat labelled diagram is required. Assume the density of water is 62.4 lb/cu. ft.
152. Water reaches the top of a rectangular dam which is 60 feet wide and 22 feet deep. Find the total force
on the dam.
153. A dam is in the form of an inverted isosceles triangle which is b feet wide and h feet deep at its deepest
point (that is, the base of the triangle is at the top). Find the total force on the dam when the lake is
full.
154. Find the total force on any dam full of alcohol in the form of an inverted triangle of base b and altitude
(maximum depth) h. Let d equal the density of alcohol.
155. A dam contains a submerged rectangular gate, 5 feet wide and 4 feet high, and the water level is 30 feet
above the top of the gate. Find the total force on the gate.
156. A dam has a submerged circular gate, 4 feet in diameter. The center of the gate is 15 feet below the
surface of the water. What is the total force on the gate?
157. Find the force due to water pressure on a rectangular window 12 in. wide and 6 in. high in the side of
a tank if the top of the window is 6 ft. below the surface.
[195 lb]
158. An elliptical window in an oil tank has its major axis horizontal. The major and minor axis are 2 and
1 ft., respectively, and the top of the window is 10 ft. below the surface. Find the force on the window
if the oil weighs 50 lb/cu. ft.
159. An oil tank is in the shape of a right circular cylinder of diameter 4 ft. with axis horizontal. Find the
total force on one end when the tank is half full of oil weighing 50 lb/cu. ft.
12.6 Work
185
160. A cylindrical tank 3 ft. in diameter is lying on its side. Find the total force due to water pressure on
one end of the tank if the water is 3/4 ft. deep.
161. A cylindrical tank 8 ft. in diameter is lying on its side. If it contains water to a depth of
√ 6 ft., find the
total force due to the water pressure on one end of the tank.
[8w(9 3 + 8π)/3 lb.]
In problems 162 to 167 find the total force due to fluid pressure on one side of each of the given areas. In
each problem assume the y-axis is horizontal and the positive x-axis extends downward. Also assume the
density of the fluid is δ.
162. The area bounded by y 2 = x, x = 4; surface of the fluid is at x = 0.
163. Same area as in problem 162 but surface of the fluid is at x = −1.
164. The area bounded by y 2 = −x, x = −9; fluid surface at x = −9.
√
√
165. The area bounded by 3y = x − 1, 3y = −(x − 1), x = 4; fluid surface at x = 0.
166. The area between 2y = x2 and y = 8, and below x = 2; fluid surface at x = 2.
167. The upper and lower halves (separately) of the circular area bounded by (x − 2)2 + y 2 = 4; fluid surface
at x = 0.
12.6
Work
In problems 168 to 177 a particle is moving along the x axis from a to b according to the force law F (x)
which is given. Find the work done.
168. F (x) = x3 + 2x2 + 6x − 1; a = 1, b = 2
169. F (x) = 8 + 2x − x2 ; a = 0, b = 3
170. F (x) = x/(1 + x2 )2 ; a = 1, b = 2
171. F (x) = (x3 + 2x2 + 1)(3x2 + 4); a = 0, b = 1
172. F (x) = sin x; a = 0, b = 2π
173. F (x) = sin x + 1; a = 0, b = 2π
174. F (x) = sin x + cos x; a = 0, b = 2π
175. F (x) = e−x ; a = 0, b = 1
176. F (x) = (ex − e−x )/2; a = 0 to b = 100
177. F (x) = e−x sin x; a = 0 to b = 50π
In problems 178 through 182, assume that each spring obeys Hooke’s Law (F = kx). Find the work done in
each case.
178. Natural length, 8 in.; 20 lb. force stretches spring
from 8 to 11 in.
179. Natural length 6 in.; 500 lb. stretches spring
1
4
1
2
in. Find the work done in stretching the spring
in. Find the work done in stretching it 1 in.
186
Applications of the Definite Integral
180. Natural length 10 in.; 30 lb. stretches it to 11 12 in. Find the work done in stretching it (a) from 10 to
12 in.; (b) from 12 to 14 in.
181. Natural length 6 in. to 5 in. (Hooke’s Law works for compression as well as for extension.)
182. Natural length 6 in.; 1200 lb. compresses it 12 in. Find the work done in compressing it from 6 in. to
4 12 in. What is the work required to bring the spring to 9 in. from its compressed state of 4 12 in?
183. A cable 100 ft. long and weighing 5 lb/ft is hanging from a windlass. Find the work done in winding
it up.
184. A tank full of water is in the form of a right circular cylinder of altitude 5 ft. and radius of base 3 ft.
How much work is done in pumping the water up to a level 10 ft. above the top of the tank?
185. A swimming pool full of water is in the form of a rectangular parallelepiped 5 ft. deep, 15 ft. wide, and
25 ft. long. Find the work required to pump the water up to a level 1 ft. above the surface of the pool.
186. The base of one cylindrical tank of radius 6 ft. and altitude 10 ft. is 3 ft. above the top of another tank
of the same size and shape. The lower tank is full of oil (50 lb/ft) and the upper tank is empty. How
much work is done in pumping all the oil from the lower tank into the upper through a pipe which
enters the upper tank through its base?
187. A trough full of water is 10 ft. long, and its cross section is in the shape of an isosceles triangle 2 ft.
wide across the top and 2 ft. high. How much work is done in pumping all the water out of the trough
over one of its ends?
188. A pyramid has a square base of sides a and a height h it is made of material weighing k pounds per
unit volume. Find the work done in lifting the material during construction.
189. A load of weight n is to be lifted from the bottom of a shaft h feet deep. The cable weighs w pounds
per foot. Find the work done.