Chapter 6 | Applications of Integration
669
146. y = x and y = x 2 rotated around the line x =2. 147. x = y 3 , x = 1 y , x =1, and x =2 rotated around the x -axis. 148. x = y 2 and y = x rotated around the line y =2. 149. [T] Left of x = sin( πy ), right of y = x , around the y -axis. For the following exercises, use technology to graph the region. Determine which method you think would be easiest to use to calculate the volume generated when the function is rotated around the specified axis. Then, use your chosen method to find the volume. 150. [T] y = x 2 and y =4 x rotated around the y -axis.
158. Use the method of shells to find the volume of a sphere of radius r .
159. Use the method of shells to find the volume of a cone with radius r and height h .
151. [T] y =cos( πx ), y = sin( πx ), x = 1
x = 5 4
4 , and
rotated around the y -axis. 152. [T] y = x 2 −2 x , x =2, and x =4 rotated around the y -axis. 153. [T] y = x 2 −2 x , x =2, and x =4 rotated around the x -axis. 154. [T] y =3 x 3 −2, y = x , and x =2 rotated around the x -axis. 155. [T] y =3 x 3 −2, y = x , and x =2 rotated around the y -axis.
160. Use the method of shells to find the volume of an ellipse ⎛ ⎝ x 2 / a 2 ⎞ ⎠ + ⎛ ⎝ y 2 / b 2 ⎞ ⎠ =1 rotated around the x -axis.
161. Use the method of shells to find the volume of a cylinder with radius r and height h .
156. [T] x = sin ⎛
⎞ ⎠ and x = 2 y rotated around the
⎝ πy 2
x -axis. 157. [T] x = y 2 , x = y 2 −2 y +1, and x =2 rotated around the y -axis. For the following exercises, use the method of shells to approximate the volumes of some common objects, which are pictured in accompanying figures.
162. Use the method of shells to find the volume of the donut created when the circle x 2 + y 2 =4 is rotated around the line x =4.
Made with FlippingBook - professional solution for displaying marketing and sales documents online