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Chapter 6 | Applications of Integration
6.3 EXERCISES For the following exercise, find the volume generated when the region between the two curves is rotated around the given axis. Use both the shell method and the washer method. Use technology to graph the functions and draw a typical slice by hand. 114. [T] Over the curve of y =3 x , x =0, and y =3 rotated around the y -axis. 115. [T] Under the curve of y =3 x , y =0, and x =3 rotated around the y -axis. 116. [T] Over the curve of y =3 x , y =0, and y =3 rotated around the x -axis. 117. [T] Under the curve of y =3 x , y =0, and x =3 rotated around the x -axis. 118. [T] Under the curve of y =2 x 3 , x =0, and x =2 rotated around the y -axis. 119. [T] Under the curve of y =2 x 3 , x =0, and x =2 rotated around the x -axis. For the following exercises, use shells to find the volumes of the given solids. Note that the rotated regions lie between the curve and the x -axis and are rotated around the y -axis.
129. y =5 x 3 −2 x 4 , x =0, and x =2 For the following exercises, use shells to find the volume generated by rotating the regions between the given curve and y =0 around the x -axis. 130. y = 1− x 2 , x =0, x =1 and the x -axis 131. y = x 2 , x =0, x =2 and the x -axis 132. y = x 3 2 , x =0, x =2, and the x -axis 133. y = 2 x 2 , x =1, x =2, and the x -axis
134. x = 1
, x = 1 5
, and y =0
1+ y 2
y 2 y , y =1, y =4, and the y -axis
135. x = 1+
136. x =cos y , y =0, and y = π 137. x = y 3 - 2 y 2 , x =0, x =9, and the y -axis 138. x = y +1, x =1, x =3, and the x -axis
and x = 3 y 4
139. x = 27 y 3
120. y =1− x 2 , x =0, and x =1 121. y =5 x 3 , x =0, and x =1 122. y = 1 x , x =1, and x =100 123. y = 1− x 2 , x =0, and x =1
For the following exercises, find the volume generated when the region between the curves is rotated around the given axis. 140. y =3− x , y =0, x =0, and x =2 rotated around the y -axis.
y = x 3 , x =0, and y =8 rotated around the
141.
124. y = 1
, x =0, and x =3
y -axis. 142. y = x 2 , y = x , rotated around the y -axis. 143. y = x , y =0, and x =1 rotated around the line x =2.
1+ x 2
125. y = sin x 2 , x =0, and x = π
126. y = 1
, x =0, and x = 1 2
1− x 2
127. y = x , x =0, and x =1
144. y = 1
4− x ,
x =1, and x =2 rotated around the
line x =4. 145. y = x and y = x 2 rotated around the y -axis.
3 , x =0, and x =1
⎛ ⎝ 1+ x 2
⎞ ⎠
128. y =
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