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.

  • A32𝜋15
  • B2𝜋3
  • C𝜋3
  • D7𝜋10
  • E64𝜋15

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.

  • A45𝜋7 volume units
  • B4514 volume units
  • C45𝜋14 volume units
  • D457 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
  • B910 volume units
  • C9𝜋10 volume units
  • D117𝜋 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.

  • A317𝜋14
  • B317𝜋7
  • C11𝜋4
  • D240𝜋7
  • E120𝜋7

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.

  • A98𝜋15 cubic units
  • B196𝜋15 cubic units
  • C637𝜋4 cubic units
  • D637𝜋2 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.

  • A𝜋7
  • B𝜋3
  • C8𝜋21
  • D𝜋4
  • E4𝜋21

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
  • B18𝜋 cubic units
  • C72 cubic units
  • D72𝜋 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.

  • A𝜋768 cubic units
  • B𝜋1,536 cubic units
  • C5𝜋768 cubic units
  • D𝜋384 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.

  • A83 cubic units
  • B8𝜋3 cubic units
  • C𝜋3 cubic units
  • D𝜋 cubic units
  • E13 cubic units

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