Chapter 6 | Applications of Integration
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72. The base is the region enclosed by y = x 2 and y =9. Slices perpendicular to the x -axis are right isosceles triangles. The intersection of one of these slices and the base is the leg of the triangle. 73. The base is the area between y = x and y = x 2 . Slices perpendicular to the x -axis are semicircles. For the following exercises, draw the region bounded by the curves. Then, use the disk method to find the volume when the region is rotated around the x -axis. 74. x + y =8, x =0, and y =0
rotated around the x -axis. 90. y = x +2, y = x +6, x =0, and x =5 91. y = x 2 and y = x +2 92. x 2 = y 3 and x 3 = y 2 93. y =4− x 2 and y =2− x 94. [T] y =cos x , y = e − x , x =0, and x =1.2927 95. y = x and y = x 2 96. y = sin x , y =5sin x , x =0and x = π 97. y = 1+ x 2 and y = 4− x 2 For the following exercises, 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. 98. y = x , x =4, and y =0 99. y = x +2, y =2 x −1, and x =0 100. y = x 3 and y = x 3 101. x = e 2 y , x = y 2 , y =0, and y = ln(2) 102. x = 9− y 2 , x = e − y , y =0, and y =3
75. y =2 x 2 , x =0, x =4, and y =0 76. y = e x +1, x =0, x =1, and y =0 77. y = x 4 , x =0, and y =1 78. y = x , x =0, x =4, and y =0 79. y = sin x , y =cos x , and x =0
80. y = 1 x , x =2, and y =3 81. x 2 − y 2 =9and x + y =9, y =0and x =0 For the following exercises, draw the region bounded by the curves. Then, find the volume when the region is rotated around the y -axis. 82. y =4− 1 2 x , x =0, and y =0 83. y =2 x 3 , x =0, x =1, and y =0 84. y =3 x 2 , x =0, and y =3 85. y = 4− x 2 , y =0, and x =0
86. y = 1
, x =0, and x =3
x +1
87. x = sec( y ) and y = π 4 ,
y =0and x =0
88. y = 1
x =0, and x =2
x +1 ,
89. y =4− x , y = x , and x =0 For the following exercises, draw the region bounded by the curves. Then, find the volume when the region is
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