121616949-math.241 - 9.11 Differential equations 227...

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9.11 Differential equations 227 Exercises 1. Compute the area of the surface formed when f ( x ) = 2 1 - x between - 1 and 0 is rotated around the x -axis. 2. Compute the surface area of example 9.39 by rotating f ( x ) = x around the x -axis. 3. Compute the area of the surface formed when f ( x ) = x 3 between 1 and 3 is rotated around the x -axis. 4. Consider the surface obtained by rotating the graph of f ( x ) = 1 /x , x 1, around the x -axis. This surface is called Gabriel’s horn or Toricelli’s trumpet . Show that Gabriel’s horn has finite volume and infinite surface area. 5. Consider the circle ( x - 2) 2 + y 2 = 1. Sketch the surface obtained by rotating this circle about the y -axis. (The surface is called a torus .) What is the surface area? 6. Consider the ellipse with equation x 2 / 4 + y 2 = 1. If the ellipse is rotated around the x -axis it forms an ellipsoid . Compute the surface area. 7. Generalize the preceding result: rotate the ellipse given by x 2 /a 2 + y 2 /b 2 = 1 about the x -axis and find the surface area of the resulting ellipsoid.
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