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.

  • A1615
  • B815
  • C23
  • D815𝜋
  • E23𝜋

Q2:

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

  • A25615
  • B163𝜋
  • C25615𝜋
  • D643
  • E163

Q3:

Use the slicing method to find the volume of the solid of revolution bounded by the graphs of 𝑓(𝑥)=𝑥2𝑥+3, 𝑥=0, and 𝑥=3 and rotated about the 𝑥-axis.

  • A452𝜋
  • B1535
  • C1535𝜋
  • D9𝜋
  • E63

Q4:

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

  • A16𝜋
  • B415𝜋
  • C215𝜋
  • D215
  • E16

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 𝑥𝑦-plane are semicircles.

  • A83
  • B23𝜋
  • C13𝜋
  • D13
  • E83𝜋

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-axis are squares.

  • A163
  • B163𝜋
  • C83𝜋
  • D83
  • E43

Q7:

Use the slicing method to find the volume of the solid whose base is the elliptical region with boundary curve 25𝑥+4𝑦=100 and cross sections perpendicular to the 𝑥-axis are isosceles right triangles with the hypotenuse in the base.

  • A2003
  • B803
  • C1003
  • D2003𝜋
  • E1003𝜋

Q8:

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

  • A723
  • B363
  • C1443𝜋
  • D1443
  • E36𝜋

Q10:

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

  • A56
  • B56𝜋
  • C524
  • D524𝜋
  • E53𝜋

Q11:

Using the slicing method, find the volume of the solid whose base is bounded by the curves 𝑧=0 and 𝑧=𝑥sin, between the first two intersections of the curves for 𝑥0, and whose perpendicular cross sections to the 𝑥𝑧-plane are squares.

  • A𝜋2
  • B2
  • C1
  • D𝜋
  • E𝜋4

Q12:

Find the volume of the solid of revolution generated by rotating the area bounded by 𝑧=𝑦sin and 𝑧=0 about the 𝑦-axis between the first two intersections of the curve and the line for 𝑥0.

  • A𝜋
  • B𝜋
  • C2𝜋
  • D𝜋4
  • E𝜋2

Q13:

Use the slicing method to find the volume of the solid whose base is bounded by the curves 𝑦=𝑥 and 𝑦=𝑥+2 and whose cross sections parallel to the 𝑦𝑧-plane are semicircles.

  • A9𝜋2
  • B81𝜋10
  • C81𝜋80
  • D9𝜋4
  • E81𝜋20

Q14:

Find the volume of the solid of revolution generated by rotating the area bounded by 𝑦=2 and 𝑦=𝑥 for 𝑥0 about the 𝑦-axis.

  • A2𝜋3
  • B𝜋
  • C8𝜋3
  • D4𝜋3
  • E2𝜋

Q15:

Find the volume of the solid of revolution generated by rotating the area bounded by 𝑧=𝑦+2 and 𝑧=𝑦 for 𝑦0 about the 𝑧-axis.

  • A6𝜋5
  • B3𝜋2
  • C𝜋6
  • D5𝜋6
  • E𝜋3

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