Subtract the curved core from a cylinder
C: . S is bounded by C, the -axis and . Find the volume when S is rotated completely about the -axis. marks
[2019/EJC/Prelim/I/6(iii)]
Solution
Volume of revolution
The curve has equation . The region is bounded by , the -axis and the line . Find the volume of the solid formed when is rotated completely about the -axis.
Hint
Find where meets . That gives the right-hand end of .
Spun about the -axis, makes a cylinder (from ) with a hollow (from ).
Volume = cylinder − the solid made by rotating the area under .
The volume under a curve about the -axis is . Using gives a number with the wrong units.
That is only the second term. is the region between and , so you must subtract it from the cylinder.
Both roots satisfy the equation, but lies in and the other root is across the vertical asymptote.
Answer · V
cubic units (3 s.f.).
Finite source region 0≤x≤1; the other intersection lies beyond the pole.
Find the right-hand end
Multiply out: , so . The root lies on the other side of the asymptote and is rejected.
Picture
At , is at , below the line . By the curve has risen to . So is the thin region between (below) and (above) for .
Cylinder minus the solid under
The rectangle up to spins into a cylinder of radius 4.5 and length 1. Removing the solid under leaves the volume of .
Evaluate
Enter the complete cylinder-minus-integral expression in the calculator, keeping the squared curve and limits 0 to 1. The source screen gives 17.51610613; round only the final result to 17.5 (3 s.f.).
Exact value (optional)
Dividing gives , so . Integrating from 0 to 1 gives and .
Explore the idea