Worksheet: Volumes by Slicing

In this worksheet, we will practice using integration to find the volume of a solid with a variable cross section.

Q1:

Use the slicing method to find the volume of the solid whose base is the region enclosed by 𝑦 = 𝑥 and 𝑦 = 1 , and whose slices perpendicular to the 𝑥 -axis are right isosceles triangles with one side parallel to the 𝑧 -axis.

  • A 2 3 𝜋
  • B 2 3
  • C 8 1 5
  • D 8 1 5 𝜋
  • E 1 6 1 5

Q2:

Use the slicing method to find the volume of the solid whose base is the region under the parabola 𝑦 = 4 𝑥 in the first quadrant and whose slices perpendicular to the 𝑥 𝑦 -plane are squares.

  • A 2 5 6 1 5 𝜋
  • B 6 4 3
  • C 1 6 3 𝜋
  • D 1 6 3
  • E 2 5 6 1 5

Q3:

Use the slicing method to find the volume of the solid of revolution bounded by the graphs of 𝑓 ( 𝑥 ) = 𝑥 2 𝑥 + 3 , 𝑥 = 0 , and 𝑥 = 3 and rotated about the 𝑥 - a x i s .

  • A 1 5 3 5 𝜋
  • B63
  • C 4 5 2 𝜋
  • D 9 𝜋
  • E 1 5 3 5

Q4:

Use the slicing method to find the volume of the solid whose base is the area between 𝑦 = 2 𝑥 and 𝑦 = 𝑥 and whose slices perpendicular to the 𝑥 - a x i s are semicircles.

  • A 1 6
  • B 4 1 5 𝜋
  • C 2 1 5
  • D 1 6 𝜋
  • E 2 1 5 𝜋

Q5:

Use the slicing method to find the volume of the solid whose base is a triangle with vertices ( 0 , 0 ) , ( 2 , 0 ) , and ( 0 , 2 ) and whose slices perpendicular to the 𝑥 𝑦 -plane are semicircles.

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

Q6:

Use the slicing method to find the volume of the solid whose base is the region inside the circle 𝑥 + 𝑦 = 1 if cross sections taken perpendicular to the 𝑦 - a x i s are squares.

  • A 8 3
  • B 1 6 3 𝜋
  • C 4 3
  • D 1 6 3
  • E 8 3 𝜋

Q7:

Use the slicing method to find the volume of the solid whose base is the elliptical region with boundary curve 2 5 𝑥 + 4 𝑦 = 1 0 0 and cross sections perpendicular to the 𝑥 - a x i s are isosceles right triangles with the hypotenuse in the base.

  • A 2 0 0 3 𝜋
  • B 1 0 0 3
  • C 1 0 0 3 𝜋
  • D 8 0 3
  • E 2 0 0 3

Q8:

Use the slicing method to find the volume of the solid whose base is the region under the parabola 𝑦 = 9 𝑥 and above the 𝑥 - a x i s and whose slices perpendicular to the 𝑦 - a x i s are squares.

Q9:

Use the slicing method to find the volume of the solid whose base is the region inside the circle with radius 3 if the cross sections taken parallel to one of the diameters are equilateral triangles.

  • A 7 2 3
  • B 3 6 3
  • C 1 4 4 3
  • D 3 6 𝜋
  • E 1 4 4 3 𝜋

Q10:

Use the slicing method to find the volume of the solid whose base is the region bounded by the lines 𝑥 + 5 𝑦 = 5 , 𝑥 = 0 , and 𝑦 = 0 if the cross sections taken perpendicular to the 𝑥 - a x i s are semicircles.

  • A 5 2 4
  • B 5 2 4 𝜋
  • C 5 6 𝜋
  • D 5 3 𝜋
  • E 5 6

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