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Worksheet: Disc Method for Rotating around a Vertical

Q1:

The region bounded by the curves π‘₯ = 3 √ 𝑦 , π‘₯ = 0 , and 𝑦 = 3 is rotated about the 𝑦 -axis. Find the volume of the resulting solid.

  • A 8 1 πœ‹
  • B 8 1 2
  • C81
  • D 8 1 πœ‹ 2
  • E 8 1 πœ‹ 4

Q2:

The region bounded by the curves π‘₯ = 2 √ 𝑦 , π‘₯ = 0 , and 𝑦 = 8 is rotated about the 𝑦 -axis. Find the volume of the resulting solid.

  • A 2 5 6 πœ‹
  • B128
  • C256
  • D 1 2 8 πœ‹
  • E 6 4 πœ‹

Q3:

Find the volume of the solid generated by turning, through a complete revolution about the 𝑦 -axis, the region bounded by the curve 9 π‘₯ βˆ’ 𝑦 = 0 and the lines π‘₯ = 0 , 𝑦 = βˆ’ 9 , and 𝑦 = 0 .

  • A 2 7 πœ‹ cubic units
  • B3 cubic units
  • C27 cubic units
  • D 3 πœ‹ cubic units

Q4:

Find the volume of the solid generated by turning, through a complete revolution about the 𝑦 -axis, the region bounded by the curve 3 π‘₯ + 8 𝑦 = 0 and the lines π‘₯ = 0 , 𝑦 = βˆ’ 1 , and 𝑦 = 0 .

  • A 6 4 πœ‹ 9 cubic units
  • B 6 4 2 7 cubic units
  • C 6 4 9 cubic units
  • D 6 4 πœ‹ 2 7 cubic units

Q5:

Find the volume of the solid generated by turning the region bounded by the curve 𝑦 = 6 π‘₯ 2 , the 𝑦 -axis, and the lines 𝑦 = 3 and 𝑦 = 4 through a complete revolution about the 𝑦 -axis.

  • A74 cubic units
  • B 7 1 2 cubic units
  • C 7 4 πœ‹ cubic units
  • D 7 πœ‹ 1 2 cubic units

Q6:

Let π‘Ž and 𝑏 be constants. Find the volume of the solid of revolution produced on turning the region bounded by the curve βˆ’ 2 𝑦 𝑏 + π‘₯ π‘Ž = 1 2 2 2 2 and the π‘₯ -axis about the 𝑦 -axis.

  • A βˆ’ 2 π‘Ž 3 𝑏 2
  • B βˆ’ πœ‹ π‘Ž 3 𝑏 2
  • C βˆ’ π‘Ž 𝑏 3 2
  • D βˆ’ 2 πœ‹ 3 π‘Ž 𝑏 2

Q7:

Let π‘Ž and 𝑏 be constants. Find the volume of the solid of revolution produced on turning the region bounded by the curve βˆ’ 7 𝑦 𝑏 + π‘₯ π‘Ž = 1 2 2 2 2 and the π‘₯ -axis about the 𝑦 -axis.

  • A βˆ’ 4 π‘Ž 2 1 𝑏 2
  • B βˆ’ 2 πœ‹ 2 1 π‘Ž 𝑏 2
  • C βˆ’ 2 π‘Ž 2 1 𝑏 2
  • D βˆ’ 4 πœ‹ 2 1 π‘Ž 𝑏 2

Q8:

Find the volume of the solid generated by revolving the region bounded by the curve 𝑦 = 8 π‘₯ and the straight lines 𝑦 = βˆ’ 4 and π‘₯ = 0 a complete revolution about the 𝑦 -axis.

  • A 8 3 cubic units
  • B 1 3 cubic units
  • C 8 πœ‹ 3 cubic units
  • D πœ‹ 3 cubic units
  • E πœ‹ cubic units

Q9:

Find the volume of the solid generated by revolving the region bounded by the curve 𝑦 = 6 π‘₯ and the straight lines 𝑦 = βˆ’ 3 and π‘₯ = 0 a complete revolution about the 𝑦 -axis.

  • A 3 2 cubic units
  • B 1 4 cubic units
  • C 3 πœ‹ 2 cubic units
  • D πœ‹ 4 cubic units
  • E 3 πœ‹ 4 cubic units