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Worksheet: Disc Method for Rotating around a Horizontal

Q1:

Consider the region bounded by the curve 𝑦 = 5 𝑒 βˆ’ 2 π‘₯ 2 and the lines 𝑦 = 0 , π‘₯ = βˆ’ 4 , and π‘₯ = 4 . Set up an integral for the volume of the solid obtained by rotating this region about the π‘₯ -axis.

  • A 2 5 πœ‹ ο„Έ 𝑒 π‘₯ 4 0 βˆ’ 4 π‘₯ 2 d
  • B 5 0 πœ‹ ο„Έ 𝑒 π‘₯ 4 0 βˆ’ 2 π‘₯ 4 d
  • C 1 0 πœ‹ ο„Έ 𝑒 π‘₯ 4 0 βˆ’ 2 π‘₯ 2 d
  • D 5 0 πœ‹ ο„Έ 𝑒 π‘₯ 4 0 βˆ’ 4 π‘₯ 2 d
  • E 2 5 πœ‹ ο„Έ 𝑒 π‘₯ 4 0 βˆ’ 2 π‘₯ 4 d

Q2:

Consider the region bounded by the curve 𝑦 = 4 𝑒 βˆ’ 5 π‘₯ 2 and the lines 𝑦 = 0 , π‘₯ = βˆ’ 1 , and π‘₯ = 1 . Set up an integral for the volume of the solid obtained by rotating this region about the π‘₯ -axis.

  • A 1 6 πœ‹ ο„Έ 𝑒 π‘₯ 1 0 βˆ’ 1 0 π‘₯ 2 d
  • B 3 2 πœ‹ ο„Έ 𝑒 π‘₯ 1 0 βˆ’ 5 π‘₯ 4 d
  • C 8 πœ‹ ο„Έ 𝑒 π‘₯ 1 0 βˆ’ 5 π‘₯ 2 d
  • D 3 2 πœ‹ ο„Έ 𝑒 π‘₯ 1 0 βˆ’ 1 0 π‘₯ 2 d
  • E 1 6 πœ‹ ο„Έ 𝑒 π‘₯ 1 0 βˆ’ 5 π‘₯ 4 d

Q3:

Consider the region bounded by the curve 𝑦 = 3 3 π‘₯ c o s 2 and the lines 𝑦 = 0 , π‘₯ = βˆ’ πœ‹ 6 , and π‘₯ = πœ‹ 6 . Set up an integral for the volume of the solid obtained by rotating this region about the π‘₯ -axis.

  • A 6 πœ‹ ο„Έ 3 π‘₯ π‘₯ πœ‹ 6 0 2 c o s d
  • B 9 πœ‹ ο„Έ 3 π‘₯ π‘₯ πœ‹ 6 0 4 c o s d
  • C 3 πœ‹ ο„Έ 3 π‘₯ π‘₯ πœ‹ 6 0 2 c o s d
  • D 1 8 πœ‹ ο„Έ 3 π‘₯ π‘₯ πœ‹ 6 0 4 c o s d
  • E 1 2 πœ‹ ο„Έ 3 π‘₯ π‘₯ πœ‹ 6 0 2 c o s d

Q4:

Find the volume of the solid obtained by rotating the region bounded by the curve 𝑦 = √ π‘₯ + 1 and the lines 𝑦 = 0 and π‘₯ = 4 about the π‘₯ -axis.

  • A 2 5 πœ‹
  • B 2 5 2
  • C25
  • D 2 5 πœ‹ 2
  • E 2 5 πœ‹ 4

Q5:

Consider the region bounded by the curves 𝑦 = π‘₯ + 4 , 𝑦 = 0 , π‘₯ = 0 , and π‘₯ = 3 . Determine the volume of the solid of revolution created by rotating this region about the π‘₯ -axis.

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

Q6:

Find the volume of the solid generated by revolving the region bounded by the curve 𝑦 = 1 βˆ’ π‘₯ 2 and the straight line π‘₯ = βˆ’ 4 a complete revolution about the π‘₯ -axis.

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