CalculusAdditional / Further Maths

Volume of revolution about the -axis

Find a volume using horizontal radii and -limits.

  • Write the horizontal radius in terms of .
  • for a full turn about the -axis.
  • Use -limits. Square the radius before integrating and give cubic units.

Example 1

The shaded region is rotated through about the -axis. Find its volume.

The shaded region between the labelled curves and the stated limits, rotated about the y-axis.
Square the radius and integrate along the rotation axis.Integrate the expanded square.Substitute the limits.

Example 2

The shaded region is rotated through about the -axis. Find its volume.

The shaded region between the labelled curves and the stated limits, rotated about the y-axis.
Square the radius and integrate along the rotation axis.Integrate the expanded square.Substitute the limits.

Your turn

Question 1

The shaded region is rotated through about the -axis. Find its volume.

The shaded region between the labelled curves and the stated limits, rotated about the y-axis.
Check answer
Square the radius and integrate along the rotation axis.Integrate the expanded square.Substitute the limits.

Volume is measured in cubic units.

Question 2

The shaded region is rotated through about the -axis. Find its volume.

The shaded region between the labelled curves and the stated limits, rotated about the y-axis.
Check answer
Square the radius and integrate along the rotation axis.Integrate the expanded square.Substitute the limits.

Volume is measured in cubic units.

Question 3

The shaded region is rotated through about the -axis. Find its volume.

The shaded region between the labelled curves and the stated limits, rotated about the y-axis.
Check answer
Square the radius and integrate along the rotation axis.Integrate the expanded square.Substitute the limits.

Volume is measured in cubic units.

Question 4

The shaded region is rotated through about the -axis. Find its volume.

The shaded region between the labelled curves and the stated limits, rotated about the y-axis.
Check answer
Square the radius and integrate along the rotation axis.Integrate the expanded square.Substitute the limits.

Volume is measured in cubic units.

Question 5

The shaded region is rotated through about the -axis. Find its volume.

The shaded region between the labelled curves and the stated limits, rotated about the y-axis.
Check answer
Square the radius and integrate along the rotation axis.Integrate the expanded square.Substitute the limits.

Volume is measured in cubic units.

← All Calculus cardsReport a mistake