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.

  • A 1 8 6 𝜋
  • B93
  • C186
  • D 9 3 𝜋
  • E 3 3 𝜋 2

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
  • C25
  • D 2 5 𝜋 2
  • E 2 5 𝜋 4

Q3:

Consider the region bounded by the curve 𝑦 = 5 𝑒 − 2 𝑥 2 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 4 0 − 4 𝑥 2 d
  • B 5 0 𝜋 𝑒 𝑥 4 0 − 2 𝑥 4 d
  • C 1 0 𝜋 𝑒 𝑥 4 0 − 2 𝑥 2 d
  • D 5 0 𝜋 𝑒 𝑥 4 0 − 4 𝑥 2 d
  • E 2 5 𝜋 𝑒 𝑥 4 0 − 2 𝑥 4 d

Q4:

Consider the region bounded by the curve 𝑦 = 3 3 𝑥 c o s 2 and the lines 𝑦 = 0 , 𝑥 = − 𝜋 6 , and 𝑥 = 𝜋 6 . Set up an integral for the volume of the solid obtained by rotating this region about the 𝑥 -axis.

  • A 6 𝜋 3 𝑥 𝑥 𝜋 6 0 2 c o s d
  • B 9 𝜋 3 𝑥 𝑥 𝜋 6 0 4 c o s d
  • C 3 𝜋 3 𝑥 𝑥 𝜋 6 0 2 c o s d
  • D 1 8 𝜋 3 𝑥 𝑥 𝜋 6 0 4 c o s d
  • E 1 2 𝜋 3 𝑥 𝑥 𝜋 6 0 2 c o s d

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.

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

Q6:

Find the volume of the solid obtained by rotating the region bounded by the curves 𝑦 = 4 + 𝑥 s e c and 𝑦 = 6 about 𝑦 = 4 where 𝑥 ∈ − 𝜋 2 , 𝜋 2 . Give your answer to two decimal places.

Q7:

Consider the region bounded by the curve 𝑦 = 3 4 𝑥 c o s 2 and the lines 𝑦 = 0 , 𝑥 = − 𝜋 8 , and 𝑥 = 𝜋 8 . Set up an integral for the volume of the solid obtained by rotating that region about 𝑦 = 4 .

  • A 𝜋 9 4 𝑥 𝑥 𝜋 8 𝜋 8 − 4 c o s d
  • B 𝜋 3 4 𝑥 𝑥 𝜋 8 𝜋 8 − 2 c o s d
  • C 𝜋 4 8 4 𝑥 − 1 8 4 𝑥 𝑥 𝜋 8 𝜋 8 − 2 4 c o s c o s d
  • D 𝜋 2 4 4 𝑥 − 9 4 𝑥 𝑥 𝜋 8 𝜋 8 − 2 4 c o s c o s d
  • E 𝜋 6 4 𝑥 𝑥 𝜋 8 𝜋 8 − 2 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 − 𝑥 2 and the lines 𝑦 = 0 , 𝑥 = − 5 , and 𝑥 = 5 about 𝑦 = − 5 .

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

Q9:

Consider the region between the curves 𝑦 = 5 𝑥 2 and 𝑥 + 𝑦 = 2 2 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 = 𝑥 s i n , 𝑦 = 𝑥 c o s , 𝑥 = 𝜋 6 , and 𝑥 = 𝜋 4 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.

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

Q12:

Calculate the volume of a solid generated by rotating the region bounded by the curve 𝑦 = 4 5 𝑥 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 6 2 5 cubic units
  • C 2 𝜋 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 = 8 𝑥 2 and 𝑥 = 𝑦 2 about 𝑦 = 5 .

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

Q14:

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

  • A 8 𝜋 1 5 cubic units
  • B 3 2 𝜋 1 5 cubic units
  • C − 1 6 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 2 , the 𝑥 -axis, and the two lines 𝑥 = − 2 and 𝑥 = 1 through a complete revolution about the 𝑥 -axis.

  • A9 cubic units
  • B 1 5 3 5 cubic units
  • C 9 𝜋 cubic units
  • D 1 5 3 𝜋 5 cubic units

Q16:

Determine, to two decimal places, the volume of the solid obtained by rotating the region bounded by the curve 𝑦 = 3 𝑒 𝑥 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 9 𝑥 + 𝑦 = 9 2 2 about 𝑦 = 5 .

  • A 6 0 √ 1 − 𝑥 𝑥 1 − 1 2 d
  • B 3 0 𝜋 √ 1 − 𝑥 𝑥 1 − 1 2 d
  • C 1 5 𝜋 √ 1 − 𝑥 𝑥 1 − 1 2 d
  • D 6 0 𝜋 √ 1 − 𝑥 𝑥 1 − 1 2 d
  • E 3 0 √ 1 − 𝑥 𝑥 1 − 1 2 d

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.

  • A14 cubic units
  • B 2 8 3 cubic units
  • C 1 4 𝜋 cubic units
  • D 2 8 𝜋 3 cubic units

Q19:

Which of the following has a volume of 𝜋 2 5 𝑥 1 5 0 d ?

  • Aa right circular cone whose height is 15 units
  • Ba sphere whose radius length is 25 units
  • Ca sphere whose radius length is 5 units
  • Da right circular cylinder 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 = 5 𝑦 2 and 𝑥 = 2 − 𝑦 2 about 𝑥 = 3 .

Q21:

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

  • A 1 6 1 − 𝑦 4 𝑦 2 0 2 d
  • B 8 𝜋 1 − 𝑦 4 𝑦 2 0 2 d
  • C 4 𝜋 1 − 𝑦 4 𝑦 2 0 2 d
  • D 1 6 𝜋 1 − 𝑦 4 𝑦 2 0 2 d
  • E 8 1 − 𝑦 4 𝑦 2 0 2 d

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