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Worksheet: Calculating Volumes of Solids of Revolution

Q1:

Find the volume of the solid generated by revolving the region bounded by the curve 𝑦 = π‘₯ βˆ’ 1 and straight lines π‘₯ = 2 and 𝑦 = 3 a complete revolution about the π‘₯ -axis.

  • A13 cubic units
  • B 2 6 3 cubic units
  • C 1 3 πœ‹ cubic units
  • D 2 6 πœ‹ 3 cubic units
  • E 1 0 πœ‹ cubic units

Q2:

Find the volume of the solid generated by revolving the region bounded by the curve 𝑦 = π‘₯ βˆ’ 2 and straight lines π‘₯ = 1 and 𝑦 = 3 a complete revolution about the π‘₯ -axis.

  • A14 cubic units
  • B 2 8 3 cubic units
  • C 1 4 πœ‹ cubic units
  • D 2 8 πœ‹ 3 cubic units

Q3:

Determine, to two decimal places, the volume of the solid obtained by rotating the region bounded by the curve 𝑦 = 3 𝑒 π‘₯ and the lines 𝑦 = 0 , π‘₯ = βˆ’ 1 , and π‘₯ = 1 about the π‘₯ -axis.

Q4:

Determine, to two decimal places, the volume of the solid obtained by rotating the region bounded by the curve 𝑦 = 𝑒 π‘₯ and the lines 𝑦 = 0 , π‘₯ = βˆ’ 3 , and π‘₯ = 1 about the π‘₯ -axis.

Q5:

Set up an integral for the volume of the solid obtained by rotating the region bounded by the curve 9 π‘₯ + 𝑦 = 9 2 2 about 𝑦 = 5 .

  • A 6 0 ο„Έ √ 1 βˆ’ π‘₯ π‘₯ 1 βˆ’ 1 2 d
  • B 3 0 πœ‹ ο„Έ √ 1 βˆ’ π‘₯ π‘₯ 1 βˆ’ 1 2 d
  • C 1 5 πœ‹ ο„Έ √ 1 βˆ’ π‘₯ π‘₯ 1 βˆ’ 1 2 d
  • D 6 0 πœ‹ ο„Έ √ 1 βˆ’ π‘₯ π‘₯ 1 βˆ’ 1 2 d
  • E 3 0 ο„Έ √ 1 βˆ’ π‘₯ π‘₯ 1 βˆ’ 1 2 d

Q6:

Set up an integral for the volume of the solid obtained by rotating the region bounded by the curve π‘₯ + 9 𝑦 = 9 2 2 about 𝑦 = 4 .

  • A 1 6 ο„Έ ο„ž 1 βˆ’ π‘₯ 9 π‘₯ 3 βˆ’ 3 2 d
  • B 8 πœ‹ ο„Έ ο„ž 1 βˆ’ π‘₯ 9 π‘₯ 3 βˆ’ 3 2 d
  • C 4 πœ‹ ο„Έ ο„ž 1 βˆ’ π‘₯ 9 π‘₯ 3 βˆ’ 3 2 d
  • D 1 6 πœ‹ ο„Έ ο„ž 1 βˆ’ π‘₯ 9 π‘₯ 3 βˆ’ 3 2 d
  • E 8 ο„Έ ο„ž 1 βˆ’ π‘₯ 9 π‘₯ 3 βˆ’ 3 2 d

Q7:

Which of the following has a volume of πœ‹ ο„Έ 2 5 π‘₯ 1 5 0 d ?

  • Aa right circular cone whose height is 15 units
  • Ba sphere whose radius length is 25 units
  • Ca sphere whose radius length is 5 units
  • Da right circular cylinder whose height is 15 units
  • Ea right circular cylinder whose height is 5 units