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

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.

  • A 3 3 𝜋 2
  • B93
  • C 1 8 6 𝜋
  • D 9 3 𝜋
  • 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.

  • A 2 5 𝜋
  • B 2 5 2
  • C 2 5 𝜋 2
  • D 2 5 𝜋 4
  • 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.

  • A 2 5 𝜋 𝑒 𝑥 d
  • B 5 0 𝜋 𝑒 𝑥 d
  • C 2 5 𝜋 𝑒 𝑥 d
  • D 5 0 𝜋 𝑒 𝑥 d
  • E 1 0 𝜋 𝑒 𝑥 d

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.

  • A 1 8 𝜋 3 𝑥 𝑥 c o s d
  • B 9 𝜋 3 𝑥 𝑥 c o s d
  • C 6 𝜋 3 𝑥 𝑥 c o s d
  • D 3 𝜋 3 𝑥 𝑥 c o s d
  • E 1 2 𝜋 3 𝑥 𝑥 c o s d

Q5:

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

  • A 2 5 𝜋 2 cubic units
  • B 1 6 𝜋 1 5 cubic units
  • C 1 6 1 5 cubic units
  • D 2 5 2 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 𝑥𝜋2,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.

  • A 𝜋 4 8 4 𝑥 1 8 4 𝑥 𝑥 c o s c o s d
  • B 𝜋 6 4 𝑥 𝑥 c o s d
  • C 𝜋 2 4 4 𝑥 9 4 𝑥 𝑥 c o s c o s d
  • D 𝜋 3 4 𝑥 𝑥 c o s d
  • E 𝜋 9 4 𝑥 𝑥 c o s d

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.

  • A 𝜋 𝑒 2 5 𝑥 d
  • B 2 𝜋 𝑒 + 1 0 𝑒 𝑥 d
  • C 𝜋 𝑒 + 2 5 𝑥 d
  • D 𝜋 𝑒 + 2 5 𝑥 d
  • E 𝜋 𝑒 + 1 0 𝑒 𝑥 d

Q9:

Consider the region between the curves 𝑦=5𝑥 and 𝑥+𝑦=2, for 𝑦0. 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=6𝑥 and the line 𝑦=5 about the 𝑥-axis.

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

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.

  • A 3 𝜋 1 0 cubic units
  • B 2 𝜋 5 cubic units
  • C 6 2 5 cubic units
  • D 6 𝜋 2 5 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.

  • A 1 9 1 𝜋 4 8 0
  • B 3 𝜋 8 0
  • C 1 9 1 𝜋 2 4 0
  • D 3 𝜋 1 6 0
  • E 1 9 1 𝜋 9 6 0

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.

  • A 8 𝜋 1 5 cubic units
  • B 1 6 𝜋 1 5 cubic units
  • C 3 2 𝜋 1 5 cubic units
  • D 1 6 𝜋 1 5 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.

  • A 1 5 3 𝜋 5 cubic units
  • B9 cubic units
  • C 9 𝜋 cubic units
  • D 1 5 3 5 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.

  • A 3 0 1 𝑥 𝑥 d
  • B 3 0 𝜋 1 𝑥 𝑥 d
  • C 1 5 𝜋 1 𝑥 𝑥 d
  • D 6 0 𝜋 1 𝑥 𝑥 d
  • E 6 0 1 𝑥 𝑥 d

Q18:

Find the volume of the solid generated by revolving the region bounded by the curve 𝑦=𝑥2 and straight lines 𝑥=1 and 𝑦=3 a complete revolution about the 𝑥-axis.

  • A 2 8 𝜋 3 cubic units
  • B 1 4 𝜋 cubic units
  • C 2 8 3 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=2𝑦 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.

  • A 8 𝜋 1 𝑦 4 𝑦 d
  • B 1 6 1 𝑦 4 𝑦 d
  • C 1 6 𝜋 1 𝑦 4 𝑦 d
  • D 8 1 𝑦 4 𝑦 d
  • E 4 𝜋 1 𝑦 4 𝑦 d

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.

  • A 9 6 𝜋 5
  • B 6 4 𝜋 5
  • C 1 1 2 𝜋 5
  • D 1 2 8 𝜋 5
  • E 5 6 𝜋 5

Q23:

Find the volume of the solid obtained by rotating the region bounded by the curves 𝑥=65𝑦 and 𝑥=𝑦 about the 𝑦-axis.

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

Q24:

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

  • A3 cubic units
  • B 2 7 𝜋 cubic units
  • C 3 𝜋 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.

  • A 1 6 2 𝜋 5
  • B 9 𝜋 2
  • C 2 4 3 𝜋 5
  • D 8 1 𝜋
  • E 3 2 4 𝜋 5

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