Worksheet: Volumes of Solids of Revolution Using the Shell Method

In this worksheet, we will practice finding the volume of a solid generated by revolving an area around a vertical or horizontal axis using the shell method.

Q1:

Find the volume of the solid obtained by rotating the region bounded by the curves 5𝑦=𝑥, 𝑦=0, 𝑥=3, and 𝑥=4 about 𝑥=2.

  • A 3 2 𝜋 1 5
  • B 2 𝜋 3
  • C 𝜋 3
  • D 7 𝜋 1 0
  • E 6 4 𝜋 1 5

Q2:

Consider the region bounded by the curve 𝑥𝑦=4 and lines 𝑦=0, 𝑥=1, and 𝑥=2. Find the volume of the solid obtained by rotating this region about 𝑥=5. Round your answer to two decimal places.

Q3:

Consider the region in the half plane 𝑦0 bounded by the curves 𝑦=4𝑥 and 𝑥+𝑦=7. Find the volume of the solid obtained by rotating this region about 𝑦-axis. Round your answer to two decimal places.

Q4:

Calculate the volume of a solid generated by rotating the region bounded by the curve 𝑦=27𝑥 and straight lines 𝑥=1 and 𝑦=4 a complete revolution about the 𝑦-axis.

  • A 4 5 𝜋 7 volume units
  • B 4 5 1 4 volume units
  • C 4 5 𝜋 1 4 volume units
  • D 4 5 7 volume units

Q5:

Calculate the volume of a solid generated by rotating the region bounded by the curve 𝑦=5𝑥2, the 𝑦-axis, and the straight line 𝑦=1 a complete revolution about the 𝑦-axis.

  • A117 volume units
  • B 9 1 0 volume units
  • C 9 𝜋 1 0 volume units
  • D 1 1 7 𝜋 volume units

Q6:

Find the volume of the solid obtained by rotating the region bounded by the curve 𝑦=𝑥 and the lines 𝑦=1 and 𝑥=2 about the line 𝑦=1.

  • A 3 1 7 𝜋 1 4
  • B 3 1 7 𝜋 7
  • C 1 1 𝜋 4
  • D 2 4 0 𝜋 7
  • E 1 2 0 𝜋 7

Q7:

Determine the volume of the solid generated by rotating the region bounded by the curve 𝑦=7𝑥 and the line 𝑦=7𝑥 a complete revolution about the 𝑥-axis.

  • A 9 8 𝜋 1 5 cubic units
  • B 1 9 6 𝜋 1 5 cubic units
  • C 6 3 7 𝜋 4 cubic units
  • D 6 3 7 𝜋 2 cubic units

Q8:

Consider the region bounded by the curves 𝑦=𝑥 and 𝑦=𝑥, for 𝑥0. Find the volume of the solid obtained by rotating this region about the 𝑥-axis.

  • A 𝜋 7
  • B 𝜋 3
  • C 8 𝜋 2 1
  • D 𝜋 4
  • E 4 𝜋 2 1

Q9:

Find the volume of the solid generated by turning the region bounded by the curves 𝑦=4𝑥, 𝑦=8, and 𝑥=5 a complete revolution about the 𝑥-axis.

  • A18 cubic units
  • B 1 8 𝜋 cubic units
  • C72 cubic units
  • D 7 2 𝜋 cubic units

Q10:

Find the volume of the solid generated by turning the region bounded by the curves 𝑦=18𝑥, 𝑦=4, 𝑦=6, and the 𝑦-axis through a complete revolution about the 𝑥-axis.

  • A 𝜋 7 6 8 cubic units
  • B 𝜋 1 , 5 3 6 cubic units
  • C 5 𝜋 7 6 8 cubic units
  • D 𝜋 3 8 4 cubic units

Q11:

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

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

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