Worksheet: Disc Method for Rotating around a Vertical

In this worksheet, we will practice finding the volume of a solid by revolution of a two-dimensional region about the y-axis using the disk method.

Q1:

The region bounded by the curves 𝑥 = 3 𝑦 , 𝑥 = 0 , and 𝑦 = 3 is rotated about the 𝑦 -axis. Find the volume of the resulting solid.

  • A 8 1 𝜋
  • B 8 1 2
  • C81
  • D 8 1 𝜋 2
  • E 8 1 𝜋 4

Q2:

Find the volume of the solid generated by turning, through a complete revolution about the 𝑦 -axis, the region bounded by the curve 9 𝑥 𝑦 = 0 and the lines 𝑥 = 0 , 𝑦 = 9 , and 𝑦 = 0 .

  • A 2 7 𝜋 cubic units
  • B3 cubic units
  • C27 cubic units
  • D 3 𝜋 cubic units

Q3:

Find the volume of the solid generated by revolving the region bounded by the curve 𝑦 = 8 𝑥 and the straight lines 𝑦 = 4 and 𝑥 = 0 a complete revolution about the 𝑦 -axis.

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

Q4:

Find the volume of the solid generated by turning the region bounded by the curve 𝑦 = 6 𝑥 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 cubic units
  • D 7 𝜋 1 2 cubic units

Q5:

Let 𝑎 and 𝑏 be constants. Find the volume of the solid of revolution produced on turning the region bounded by the curve 2 𝑦 𝑏 + 𝑥 𝑎 = 1 2 2 2 2 and the 𝑥 -axis about the 𝑦 -axis.

  • A 2 𝑎 3 𝑏 2
  • B 𝜋 𝑎 3 𝑏 2
  • C 𝑎 𝑏 3 2
  • D 2 𝜋 3 𝑎 𝑏 2

Q6:

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

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

Q7:

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

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

Q8:

Let 𝑎 and 𝑏 be constants. Find the volume of the solid of revolution produced on turning the region bounded by the curve 7 𝑦 𝑏 + 𝑥 𝑎 = 1 2 2 2 2 and the 𝑥 -axis about the 𝑦 -axis.

  • A 4 𝑎 2 1 𝑏 2
  • B 2 𝜋 2 1 𝑎 𝑏 2
  • C 2 𝑎 2 1 𝑏 2
  • D 4 𝜋 2 1 𝑎 𝑏 2

Q9:

Find the volume of the solid generated by revolving the region bounded by the curve 𝑦 = 6 𝑥 and the straight lines 𝑦 = 3 and 𝑥 = 0 a complete revolution about the 𝑦 -axis.

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

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