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Lesson: Washer Method for Rotating around a Horizontal

Sample Question Videos

Worksheet • 14 Questions • 2 Videos

Q1:

Find the volume of the solid obtained by rotating the region bounded by the curve 𝑦 = π‘₯  and the lines 𝑦 = 1 and π‘₯ = 2 about the line 𝑦 = βˆ’ 1 .

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

Q2:

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

Q3:

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

Q4:

Find the volume of the solid obtained by rotating the region bounded by the curve 𝑦 = 6 βˆ’ π‘₯  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

Q5:

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

Q6:

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

Q7:

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

Q8:

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

Q9:

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

Q10:

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.

Q11:

Find the volume of the solid obtained by rotating the region bounded by the curves 𝑦 = 4 + π‘₯ s e c and 𝑦 = 6 about 𝑦 = 4 where π‘₯ ∈ ο€» βˆ’ πœ‹ 2 , πœ‹ 2  . Give your answer to two decimal places.

Q12:

Consider the region bounded by the curve 𝑦 = 3 4 π‘₯ c o s 2 and the lines 𝑦 = 0 , π‘₯ = βˆ’ πœ‹ 8 , and π‘₯ = πœ‹ 8 . Set up an integral for the volume of the solid obtained by rotating that region about 𝑦 = 4 .

  • A πœ‹ ο„Έ ο€Ή 2 4 4 π‘₯ βˆ’ 9 4 π‘₯  π‘₯ πœ‹ 8 πœ‹ 8 βˆ’ 2 4 c o s c o s d
  • B πœ‹ ο„Έ 6 4 π‘₯ π‘₯ πœ‹ 8 πœ‹ 8 βˆ’ 2 c o s d
  • C πœ‹ ο„Έ 3 4 π‘₯ π‘₯ πœ‹ 8 πœ‹ 8 βˆ’ 2 c o s d
  • D πœ‹ ο„Έ ο€Ή 4 8 4 π‘₯ βˆ’ 1 8 4 π‘₯  π‘₯ πœ‹ 8 πœ‹ 8 βˆ’ 2 4 c o s c o s d
  • E πœ‹ ο„Έ 9 4 π‘₯ π‘₯ πœ‹ 8 πœ‹ 8 βˆ’ 4 c o s d

Q13:

Set up an integral for the volume of the solid obtained by rotating the region bounded by the curve 𝑦 = 𝑒 βˆ’ π‘₯ 2 and the lines 𝑦 = 0 , π‘₯ = βˆ’ 5 , and π‘₯ = 5 about 𝑦 = βˆ’ 5 .

  • A πœ‹ ο„Έ ο€Ί 𝑒 + 1 0 𝑒  π‘₯ 5 βˆ’ 5 βˆ’ 2 π‘₯ βˆ’ π‘₯ 2 2 d
  • B πœ‹ ο„Έ ο€Ή 𝑒 + 2 5  π‘₯ 5 βˆ’ 5 βˆ’ 2 π‘₯ d
  • C πœ‹ ο„Έ ο€Ί 𝑒 βˆ’ 2 5  π‘₯ 5 βˆ’ 5 βˆ’ 2 π‘₯ 2 d
  • D πœ‹ ο„Έ ο€Ί 𝑒 + 2 5  π‘₯ 5 βˆ’ 5 βˆ’ 2 π‘₯ 2 d
  • E 2 πœ‹ ο„Έ ο€Ί 𝑒 + 1 0 𝑒  π‘₯ 5 βˆ’ 5 βˆ’ 2 π‘₯ βˆ’ π‘₯ 2 d

Q14:

Find the volume of the solid obtained by rotating the region bounded by the curves 𝑦 = π‘₯ s i n , 𝑦 = π‘₯ c o s , π‘₯ = πœ‹ 6 , and π‘₯ = πœ‹ 4 about 𝑦 = βˆ’ 1 . Give your answer to two decimal places.

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