Engineering Board Exam Reviewer · Applications: Volume, Arc Length, Surface Area · hard

The circle (x-5)2+y2=4 is revolved about the y-axis to form a torus. Find its volume.

  1. 20 π2 197.39
  2. 80 π2 789.57
  3. 40 π 125.66
  4. 40 π2 394.78

Answer

D. 40 π2 394.78

Solution

Using shells: V=2π 37x·2sqrt(4-(x-5)2),dx. With u=x-5: 4π -22(u+5)sqrt(4-u2),du=4π·5·2π=40π2 (the odd part integrates to 0 and -22sqrt(4-u2),du=2π). Pappus check: V=(2π·5)(π·22)=40π2.

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