Find the region before you choose the radii
Curve C: .
(i) Without a calculator, determine its exact stationary point. marks
(ii) Sketch C, stating clearly any asymptotes. marks
R is bounded by C, and the -axis.
(iii) Find its exact area. marks
(iv) Find the volume when R is rotated radians about the -axis. marks
[2021/EJC/Prelim/II/4]
Solution
If this part uses an earlier result, revisit (iii) Area between curves or open all working together.
Volume of revolution
Find the volume generated when is rotated through radians about the -axis.
Hint
Rotating about the -axis, slice horizontally, so everything must be a function of .
The outer boundary is the line. For low the inner boundary is the curve; for high it is the -axis itself.
The outer solid is a cone. Subtract the hollow left by the curve.
The printed side note says cylinder, but the line makes a cone. The working correctly uses .
Rotating about the -axis requires . Do not integrate with respect to unless you use shells.
Answer · (iv)
Volume cubic units.
Write both boundaries as
From : . lies in , so take the minus branch.
Outer solid: a cone
Spinning the line from (radius 1) up to (radius 0) makes a cone of radius 1 and height .
Outer-disc integral alternative
This is the source’s other method: integrate the discs formed by the straight-line boundary. It equals the cone formula above; subtract the same curved hollow in either method.
Subtract the hollow under the curve
For between and there is a gap between the -axis and the curve. That gap is not part of , so its volume is removed.
Evaluate with the calculator
The integral is , so .
Explore the idea
Open interactive exploration
Sketch first, then choose radii
The region touches the axis only for . Below that the curve moves away from the axis, which is what creates the hollow. Seeing that on a sketch is what tells you whether you need one integral or a subtraction.