The portal has been deactivated. Please contact your portal admin.

Lesson Worksheet: Volumes of Solids of Revolution Mathematics • Higher Education

In this worksheet, we will practice finding the volume of a solid generated by revolving a region around either a horizontal or a vertical line using integration.

Q1:

Which of the following has the volume represented by the integration πœ‹ο„Έ25π‘₯d?

  • Aa sphere whose radius length is 25 units
  • Ba sphere whose radius length is 5 units
  • Ca right circular cylinder whose height is 15 units and radius is 5 units
  • Da right circular cone whose height is 15 units and radius is 25 units
  • Ea right circular cylinder whose height is 5 units and radius is 15 units

Q2:

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.

  • A33πœ‹2
  • B93
  • C186πœ‹
  • D93πœ‹
  • E186

Q3:

Find the volume of the solid generated by turning, through a complete revolution about the 𝑦-axis, the region bounded by the curve 9π‘₯βˆ’π‘¦=0 and the lines π‘₯=0, 𝑦=βˆ’9, and 𝑦=0.

  • A3 cubic units
  • B27πœ‹ cubic units
  • C3πœ‹ cubic units
  • D27 cubic units

Q4:

Find the volume of the solid generated by rotating the region bounded by the curve 𝑦=βˆ’π‘₯+2π‘₯ and the π‘₯-axis a complete revolution about the π‘₯-axis.

  • A8πœ‹15 cubic units
  • B16πœ‹15 cubic units
  • C32πœ‹15 cubic units
  • Dβˆ’16πœ‹15 cubic units

Q5:

The region bounded by the curves π‘₯=3βˆšπ‘¦, π‘₯=0, and 𝑦=3 is rotated about the 𝑦-axis. Find the volume of the resulting solid.

  • A81
  • B81πœ‹2
  • C812
  • D81πœ‹4
  • E81πœ‹

Q6:

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.

  • A25πœ‹
  • B252
  • C25πœ‹2
  • D25πœ‹4
  • E25

Q7:

Let π‘Ž and 𝑏 be constants. Find the volume of the solid of revolution produced on turning the region bounded by the curve βˆ’2𝑦𝑏+π‘₯π‘Ž=1 and the π‘₯-axis about the 𝑦-axis.

  • Aβˆ’πœ‹π‘Ž3π‘οŠ¨
  • Bβˆ’2πœ‹3π‘Žπ‘οŠ¨
  • Cβˆ’π‘Žπ‘3
  • Dβˆ’2π‘Ž3π‘οŠ¨

Q8:

Find the volume of the solid obtained by rotating the region bounded by the curve 𝑦=π‘₯ and the line π‘₯=3𝑦 about the 𝑦-axis.

  • A162πœ‹5
  • B9πœ‹2
  • C243πœ‹5
  • D81πœ‹
  • E324πœ‹5

Q9:

Find the volume of the solid obtained by rotating the region bounded by the curves π‘₯=6βˆ’5π‘¦οŠ¨ and π‘₯=𝑦οŠͺ about the 𝑦-axis.

  • A2πœ‹9
  • B376πœ‹9
  • C124πœ‹15
  • D42πœ‹
  • E188πœ‹9

Q10:

Consider the region between the curves 𝑦=5π‘₯ and π‘₯+𝑦=2, for 𝑦β‰₯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.

This lesson includes 56 additional questions and 217 additional question variations for subscribers.

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.