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Lesson: Finding the Volume of a Solid by Rotating around the Horizontal Line Using the Washer Method

Worksheet • 14 Questions

Q1:

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

Q2:

Consider the region between the curves 𝑦 = 4 π‘₯ 2 and π‘₯ + 𝑦 = 3 2 2 , for 𝑦 β‰₯ 0 . Find the volume of the solid of revolution obtained by rotating this region about the π‘₯ -axis, giving your answer to two decimal places.

Q3:

Determine the volume of the solid generated by rotating the region bounded by the curve 𝑦 = 7 π‘₯ 2 and the line 𝑦 = 7 π‘₯ a complete revolution about the π‘₯ -axis.

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

Q4:

Determine the volume of the solid generated by rotating the region bounded by the curve 𝑦 = 4 π‘₯ 2 and the line 𝑦 = 4 π‘₯ a complete revolution about the π‘₯ -axis.

  • A 3 2 πœ‹ 1 5 cubic units
  • B 6 4 πœ‹ 1 5 cubic units
  • C 5 6 πœ‹ cubic units
  • D 2 8 πœ‹ cubic units

Q5:

Find the volume of the solid obtained by rotating the region bounded by the curve 𝑦 = 6 βˆ’ π‘₯ 2 and the line 𝑦 = 5 about the π‘₯ -axis.

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

Q6:

Find the volume of the solid obtained by rotating the region bounded by the curve 𝑦 = 6 βˆ’ π‘₯ 2 and the line 𝑦 = 2 about the π‘₯ -axis.

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

Q7:

Consider the region bounded by the curves 𝑦 = π‘₯ 3 and 𝑦 = π‘₯ , for π‘₯ β‰₯ 0 . Find the volume of the solid obtained by rotating this region about the π‘₯ -axis.

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

Q8:

Consider the region bounded by the curves 𝑦 = 8 π‘₯ 3 and 𝑦 = 8 π‘₯ , for π‘₯ β‰₯ 0 . Find the volume of the solid obtained by rotating this region about the π‘₯ -axis.

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

Q9:

Consider the region bounded by the curves 𝑦 = 5 π‘₯ 3 and 𝑦 = 5 π‘₯ , for π‘₯ β‰₯ 0 . Find the volume of the solid obtained by rotating this region about the π‘₯ -axis.

  • A 1 0 0 πœ‹ 2 1
  • B 5 πœ‹ 4
  • C 2 5 πœ‹ 7
  • D 2 5 πœ‹ 3
  • E 2 0 0 πœ‹ 2 1

Q10:

Calculate the volume of a solid generated by rotating the region bounded by the curve 𝑦 = 4 5 π‘₯ and straight lines π‘₯ = 2 , π‘₯ = 8 , and 𝑦 = 0 a complete revolution about the π‘₯ -axis.

  • A 6 πœ‹ 2 5 cubic units
  • B 6 2 5 cubic units
  • C 2 πœ‹ 5 cubic units
  • D 3 πœ‹ 1 0 cubic units

Q11:

Find the volume of the solid generated by turning the region bounded by the curves 𝑦 = 4 √ π‘₯ , 𝑦 = βˆ’ 8 , and π‘₯ = 5 a complete revolution about the π‘₯ -axis.

  • A 7 2 πœ‹ cubic units
  • B72 cubic units
  • C 1 8 πœ‹ cubic units
  • D18 cubic units

Q12:

Find the volume of the solid generated by turning the region bounded by the curves 𝑦 = 1 8 π‘₯ , 𝑦 = βˆ’ 4 , 𝑦 = βˆ’ 6 , and the 𝑦 -axis through a complete revolution about the π‘₯ -axis.

  • A πœ‹ 7 6 8 cubic units
  • B πœ‹ 3 8 4 cubic units
  • C 5 πœ‹ 7 6 8 cubic units
  • D πœ‹ 1 5 3 6 cubic units

Q13:

Find the volume of the solid generated by rotating the region bounded by the curve 𝑦 = βˆ’ π‘₯ + 2 π‘₯ 2 and the π‘₯ -axis a complete revolution about the π‘₯ -axis.

  • A 1 6 πœ‹ 1 5 cubic units
  • B 3 2 πœ‹ 1 5 cubic units
  • C βˆ’ 1 6 πœ‹ 1 5 cubic units
  • D 8 πœ‹ 1 5 cubic units

Q14:

Find the volume of the solid generated by turning the region bounded by the curve 𝑦 = π‘₯ + 2 2 , the π‘₯ -axis, and the two lines π‘₯ = βˆ’ 2 and π‘₯ = 1 through a complete revolution about the π‘₯ -axis.

  • A 1 5 3 πœ‹ 5 cubic units
  • B 1 5 3 5 cubic units
  • C 9 πœ‹ cubic units
  • D9 cubic units
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