Because there is an opening between the region and the line being revolved about, there will be a cylindrical opening after rotating.
Solid: Cylinder of r=5 and h=2, with a cylindrical opening with r=1 and h=2 Volume: 48 π units^3
Practice makes perfect
If we revolve the region about the line x=5, the solid will be a cylinder.
Because there is an opening between the region and the line being revolved about, there will be a cylindrical opening after rotating. The volume of the figure created after the revolution will be the difference of the volume of the big cylinder and the volume of the cylindrical opening.
Volume of Big Cylinder
Let's start by finding the volume of the big cylinder. The region is revolved around a vertical line, so the height will be equal to the length of the region, 2 units. The radius will be equal to the distance between the far end of the region to the line, 5 units. Let's substitute these values into the formula for the volume of a cylinder.
The volume of the big cylinder is 50 π cubic units.
Volume of Cylindrical Opening
Next let's find the volume of the cylindrical opening. The height will remain the same at 2 units. This time, the radius will be equal to the distance between the near end of the region to the line, 1 unit. Let's substitute these values into the formula for the volume of a cylinder.
The volume of the cylindrical opening is 2 π cubic units.
Volume of Solid of Revolution
Finally, to find the volume of the solid of revolution we will take the difference between the volumes of the big cylinder and cylindrical opening.
Volume of solid= V_(Big cylinder)-V_(Cylindrical opening)
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Volume of solid= 50 π-2 π=48 π units^3
The volume of the solid of revolution is 48 π cubic units.
