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Worksheet: Washer Method for Rotating around a Horizontal

Q1:

Consider the region between the curves ๐‘ฆ = 5 ๐‘ฅ 2 and ๐‘ฅ + ๐‘ฆ = 2 2 2 , for ๐‘ฆ โ‰ฅ 0 . Find the volume of the solid of revolution obtained by rotating this region about the ๐‘ฅ -axis, giving your answer to two decimal places.

Q2:

Consider the region between the curves ๐‘ฆ = 4 ๐‘ฅ 2 and ๐‘ฅ + ๐‘ฆ = 3 2 2 , for ๐‘ฆ โ‰ฅ 0 . Find the volume of the solid of revolution obtained by rotating this region about the ๐‘ฅ -axis, giving your answer to two decimal places.

Q3:

Determine the volume of the solid generated by rotating the region bounded by the curve ๐‘ฆ = 7 ๐‘ฅ 2 and the line ๐‘ฆ = 7 ๐‘ฅ a complete revolution about the ๐‘ฅ -axis.

  • A 6 3 7 ๐œ‹ 4 cubic units
  • B 1 9 6 ๐œ‹ 1 5 cubic units
  • C 6 3 7 ๐œ‹ 2 cubic units
  • D 9 8 ๐œ‹ 1 5 cubic units

Q4:

Determine the volume of the solid generated by rotating the region bounded by the curve ๐‘ฆ = 4 ๐‘ฅ 2 and the line ๐‘ฆ = 4 ๐‘ฅ a complete revolution about the ๐‘ฅ -axis.

  • A 2 8 ๐œ‹ cubic units
  • B 6 4 ๐œ‹ 1 5 cubic units
  • C 5 6 ๐œ‹ cubic units
  • D 3 2 ๐œ‹ 1 5 cubic units

Q5:

Find the volume of the solid obtained by rotating the region bounded by the curve ๐‘ฆ = 6 โˆ’ ๐‘ฅ 2 and the line ๐‘ฆ = 5 about the ๐‘ฅ -axis.

  • A 1 4 4 ๐œ‹ 5
  • B 3 2 2 ๐œ‹ 5
  • C 3 6 ๐œ‹ 5
  • D 7 2 ๐œ‹ 5
  • E 4 ๐œ‹ 3

Q6:

Find the volume of the solid obtained by rotating the region bounded by the curve ๐‘ฆ = 6 โˆ’ ๐‘ฅ 2 and the line ๐‘ฆ = 2 about the ๐‘ฅ -axis.

  • A 7 6 8 ๐œ‹ 5
  • B 4 6 4 ๐œ‹ 5
  • C 1 9 2 ๐œ‹ 5
  • D 3 8 4 ๐œ‹ 5
  • E 3 2 ๐œ‹ 3

Q7:

Consider the region bounded by the curves ๐‘ฆ = ๐‘ฅ 3 and ๐‘ฆ = ๐‘ฅ , for ๐‘ฅ โ‰ฅ 0 . Find the volume of the solid obtained by rotating this region about the ๐‘ฅ -axis.

  • A 8 ๐œ‹ 2 1
  • B ๐œ‹ 7
  • C ๐œ‹ 3
  • D 4 ๐œ‹ 2 1
  • E ๐œ‹ 4

Q8:

Consider the region bounded by the curves ๐‘ฆ = 8 ๐‘ฅ 3 and ๐‘ฆ = 8 ๐‘ฅ , for ๐‘ฅ โ‰ฅ 0 . Find the volume of the solid obtained by rotating this region about the ๐‘ฅ -axis.

  • A 5 1 2 ๐œ‹ 2 1
  • B 6 4 ๐œ‹ 7
  • C 6 4 ๐œ‹ 3
  • D 2 5 6 ๐œ‹ 2 1
  • E 2 ๐œ‹

Q9:

Consider the region bounded by the curves ๐‘ฆ = 5 ๐‘ฅ 3 and ๐‘ฆ = 5 ๐‘ฅ , for ๐‘ฅ โ‰ฅ 0 . Find the volume of the solid obtained by rotating this region about the ๐‘ฅ -axis.

  • A 2 0 0 ๐œ‹ 2 1
  • B 2 5 ๐œ‹ 7
  • C 2 5 ๐œ‹ 3
  • D 1 0 0 ๐œ‹ 2 1
  • E 5 ๐œ‹ 4

Q10:

Calculate the volume of a solid generated by rotating the region bounded by the curve ๐‘ฆ = 4 5 ๐‘ฅ and straight lines ๐‘ฅ = 2 , ๐‘ฅ = 8 , and ๐‘ฆ = 0 a complete revolution about the ๐‘ฅ -axis.

  • A 3 ๐œ‹ 1 0 cubic units
  • B 6 2 5 cubic units
  • C 2 ๐œ‹ 5 cubic units
  • D 6 ๐œ‹ 2 5 cubic units

Q11:

Find the volume of the solid generated by turning the region bounded by the curves ๐‘ฆ = 4 โˆš ๐‘ฅ , ๐‘ฆ = โˆ’ 8 , and ๐‘ฅ = 5 a complete revolution about the ๐‘ฅ -axis.

  • A18 cubic units
  • B72 cubic units
  • C 1 8 ๐œ‹ cubic units
  • D 7 2 ๐œ‹ cubic units

Q12:

Find the volume of the solid generated by turning the region bounded by the curves ๐‘ฆ = 1 8 ๐‘ฅ , ๐‘ฆ = โˆ’ 4 , ๐‘ฆ = โˆ’ 6 , and the ๐‘ฆ -axis through a complete revolution about the ๐‘ฅ -axis.

  • A ๐œ‹ 1 5 3 6 cubic units
  • B ๐œ‹ 3 8 4 cubic units
  • C 5 ๐œ‹ 7 6 8 cubic units
  • D ๐œ‹ 7 6 8 cubic units

Q13:

Find the volume of the solid generated by rotating the region bounded by the curve ๐‘ฆ = โˆ’ ๐‘ฅ + 2 ๐‘ฅ 2 and the ๐‘ฅ -axis a complete revolution about the ๐‘ฅ -axis.

  • A 8 ๐œ‹ 1 5 cubic units
  • B 3 2 ๐œ‹ 1 5 cubic units
  • C โˆ’ 1 6 ๐œ‹ 1 5 cubic units
  • D 1 6 ๐œ‹ 1 5 cubic units

Q14:

Find the volume of the solid generated by turning the region bounded by the curve ๐‘ฆ = ๐‘ฅ + 2 2 , the ๐‘ฅ -axis, and the two lines ๐‘ฅ = โˆ’ 2 and ๐‘ฅ = 1 through a complete revolution about the ๐‘ฅ -axis.

  • A9 cubic units
  • B 1 5 3 5 cubic units
  • C 9 ๐œ‹ cubic units
  • D 1 5 3 ๐œ‹ 5 cubic units