CalculusAdditional / Further Maths

Volume of revolution about the -axis

Find a volume by squaring the vertical radius and integrating.

  • Rotate the region between and the -axis through .
  • . Square the whole radius before integrating.
  • Use limits measured along the -axis and give the volume in 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 x-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 x-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 x-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 x-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 x-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 x-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 x-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.

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