Worksheet: Washer Method for Rotating around a Horizontal

In this worksheet, we will practice finding the volume of the solid of revolution using the washer method around a horizontal line.

Q1:

Find the volume of the solid obtained by rotating the region bounded by the curve 𝑦 = 𝑥 and the lines 𝑦 = 1 and 𝑥 = 2 about the line 𝑦 = 1 .

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

Q2:

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

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

Q3:

Consider the region bounded by the curves 𝑦 = 𝑥 3 and 𝑦 = 𝑥 , for 𝑥 0 . Find the volume of the solid obtained by rotating this region about the 𝑥 -axis.

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

Q4:

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

Q5:

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

Q6:

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

Q7:

Find the volume of the solid generated by turning the region bounded by the curves 𝑦 = 4 𝑥 , 𝑦 = 8 , and 𝑥 = 5 a complete revolution about the 𝑥 -axis.

  • A18 cubic units
  • B72 cubic units
  • C 1 8 𝜋 cubic units
  • D 7 2 𝜋 cubic units

Q8:

Find the volume of the solid generated by turning the region bounded by the curves 𝑦 = 1 8 𝑥 , 𝑦 = 4 , 𝑦 = 6 , and the 𝑦 -axis through a complete revolution about the 𝑥 -axis.

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

Q9:

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

Q10:

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.

Q11:

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.

Q12:

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

Q13:

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

Q14:

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.

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