Worksheet: Volumes of Solids of Revolution Using the Disk and Washer Methods

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In this worksheet, we will practice finding the volume of a solid generated by revolving a region around either a horizontal or a vertical line using the disk and washer methods.

Q1:

Consider the region bounded by the curves 𝑦=𝑥+4, 𝑦=0, 𝑥=0, and 𝑥=3. Determine the volume of the solid of revolution created by rotating this region about the 𝑥-axis.

  • A33𝜋2
  • B93
  • C186𝜋
  • D93𝜋
  • E186

Q2:

Find the volume of the solid obtained by rotating the region bounded by the curve 𝑦=√𝑥+1 and the lines 𝑦=0 and 𝑥=4 about the 𝑥-axis.

  • A25𝜋
  • B252
  • C25𝜋2
  • D25𝜋4
  • E25

Q3:

Consider the region bounded by the curve 𝑦=5𝑒 and the lines 𝑦=0, 𝑥=−4, and 𝑥=4. Set up an integral for the volume of the solid obtained by rotating this region about the 𝑥-axis.

  • A25𝜋𝑒𝑥d
  • B50𝜋𝑒𝑥d
  • C25𝜋𝑒𝑥d
  • D50𝜋𝑒𝑥d
  • E10𝜋𝑒𝑥d

Q4:

Consider the region bounded by the curve 𝑦=33𝑥cos and the lines 𝑦=0, 𝑥=−𝜋6, and 𝑥=𝜋6. Set up an integral for the volume of the solid obtained by rotating this region about the 𝑥-axis.

  • A18𝜋3𝑥𝑥cosd
  • B9𝜋3𝑥𝑥cosd
  • C6𝜋3𝑥𝑥cosd
  • D3𝜋3𝑥𝑥cosd
  • E12𝜋3𝑥𝑥cosd

Q5:

Find the volume of the solid generated by revolving the region bounded by the curve 𝑦=1−𝑥 and the straight line 𝑥=−4 a complete revolution about the 𝑥-axis.

  • A25𝜋2 cubic units
  • B16𝜋15 cubic units
  • C1615 cubic units
  • D252 cubic units

Q6:

Find the volume of the solid obtained by rotating the region bounded by the curves 𝑦=4+𝑥sec and 𝑦=6 about 𝑦=4 where 𝑥∈−𝜋2,𝜋2. Give your answer to two decimal places.

Q7:

Consider the region bounded by the curve 𝑦=34𝑥cos and the lines 𝑦=0, 𝑥=−𝜋8, and 𝑥=𝜋8. Set up an integral for the volume of the solid obtained by rotating that region about 𝑦=4.

  • A𝜋484𝑥−184𝑥𝑥coscosd
  • B𝜋64𝑥𝑥cosd
  • C𝜋244𝑥−94𝑥𝑥coscosd
  • D𝜋34𝑥𝑥cosd
  • E𝜋94𝑥𝑥cosd

Q8:

Set up an integral for the volume of the solid obtained by rotating the region bounded by the curve 𝑦=𝑒 and the lines 𝑦=0, 𝑥=−5, and 𝑥=5 about 𝑦=−5.

  • A𝜋𝑒−25𝑥d
  • B2𝜋𝑒+10𝑒𝑥d
  • C𝜋𝑒+25𝑥d
  • D𝜋𝑒+25𝑥d
  • E𝜋𝑒+10𝑒𝑥d

Q9:

Consider the region between the curves 𝑦=5𝑥 and 𝑥+𝑦=2, for 𝑦≥0. Find the volume of the solid of revolution obtained by rotating this region about the 𝑥-axis, giving your answer to two decimal places.

Q10:

Find the volume of the solid obtained by rotating the region bounded by the curves 𝑦=𝑥sin, 𝑦=𝑥cos, 𝑥=𝜋6, and 𝑥=𝜋4 about 𝑦=−1. Give your answer to two decimal places.

Q11:

Find the volume of the solid obtained by rotating the region bounded by the curve 𝑦=6−𝑥 and the line 𝑦=5 about the 𝑥-axis.

  • A72𝜋5
  • B36𝜋5
  • C4𝜋3
  • D144𝜋5
  • E322𝜋5

Q12:

Calculate the volume of a solid generated by rotating the region bounded by the curve 𝑦=45𝑥 and straight lines 𝑥=2, 𝑥=8, and 𝑦=0 a complete revolution about the 𝑥-axis.

  • A3𝜋10 cubic units
  • B2𝜋5 cubic units
  • C625 cubic units
  • D6𝜋25 cubic units

Q13:

Find the volume of the solid obtained by rotating the region bounded by the curves 𝑦=8𝑥 and 𝑥=𝑦 about 𝑦=5.

  • A191𝜋480
  • B3𝜋80
  • C191𝜋240
  • D3𝜋160
  • E191𝜋960

Q14:

Find the volume of the solid generated by rotating the region bounded by the curve 𝑦=−𝑥+2𝑥 and the 𝑥-axis a complete revolution about the 𝑥-axis.

  • A8𝜋15 cubic units
  • B16𝜋15 cubic units
  • C32𝜋15 cubic units
  • D−16𝜋15 cubic units

Q15:

Find the volume of the solid generated by turning the region bounded by the curve 𝑦=𝑥+2, the 𝑥-axis, and the two lines 𝑥=−2 and 𝑥=1 through a complete revolution about the 𝑥-axis.

  • A153𝜋5 cubic units
  • B9 cubic units
  • C9𝜋 cubic units
  • D1535 cubic units

Q16:

Determine, to two decimal places, the volume of the solid obtained by rotating the region bounded by the curve 𝑦=3𝑒 and the lines 𝑦=0, 𝑥=−1, and 𝑥=1 about the 𝑥-axis.

Q17:

Set up an integral for the volume of the solid obtained by rotating the region bounded by the curve 9𝑥+𝑦=9 about 𝑦=5.

  • A30√1−𝑥𝑥d
  • B30𝜋√1−𝑥𝑥d
  • C15𝜋√1−𝑥𝑥d
  • D60𝜋√1−𝑥𝑥d
  • E60√1−𝑥𝑥d

Q18:

Find the volume of the solid generated by revolving the region bounded by the curve 𝑦=𝑥−2 and straight lines 𝑥=1 and 𝑦=3 a complete revolution about the 𝑥-axis.

  • A28𝜋3 cubic units
  • B14𝜋 cubic units
  • C283 cubic units
  • D14 cubic units

Q19:

Which of the following has a volume of 𝜋25𝑥d?

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

Q20:

Determine, to two decimal places, the volume of the solid obtained by rotating the region bounded by the curves 𝑥=5𝑦 and 𝑥=2−𝑦 about 𝑥=3.

Q21:

Set up an integral for the volume of the solid obtained by rotating the region bounded by the curve 4𝑥+𝑦=4 about 𝑥=2.

  • A8𝜋1−𝑦4𝑦d
  • B161−𝑦4𝑦d
  • C16𝜋1−𝑦4𝑦d
  • D81−𝑦4𝑦d
  • E4𝜋1−𝑦4𝑦d

Q22:

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

  • A96𝜋5
  • B64𝜋5
  • C112𝜋5
  • D128𝜋5
  • E56𝜋5

Q23:

Find the volume of the solid obtained by rotating the region bounded by the curves 𝑥=6−5𝑦 and 𝑥=𝑦 about the 𝑦-axis.

  • A2𝜋9
  • B376𝜋9
  • C124𝜋15
  • D42𝜋
  • E188𝜋9

Q24:

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

  • A3 cubic units
  • B27𝜋 cubic units
  • C3𝜋 cubic units
  • D27 cubic units

Q25:

Find the volume of the solid obtained by rotating the region bounded by the curve 𝑦=𝑥 and the line 𝑥=3𝑦 about the 𝑦-axis.

  • A162𝜋5
  • B9𝜋2
  • C243𝜋5
  • D81𝜋
  • E324𝜋5

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