Q159 · Practice questionTechnology-freeComplex familiar6 marks
QUESTION 159 (6 marks)
The region is bounded by and for , as shown. Two solids are formed by rotating through one complete revolution: one about the -axis, the other about the -axis.
a)[3 marks]
Determine the exact volume of the solid formed about the -axis.
b)[3 marks]
Determine the exact volume of the solid formed about the -axis, and calculate .
WORKED SOLUTION
6 marksPractice marking scheme
ANSWER
(a) . (b) ; .
Worked solution
(a) For a vertical slice the outer radius is and the inner radius is . Therefore
(b) Rewrite the boundaries as and . For , the outer radius is and the inner radius is . Thus
Consequently . The same planar region produces different volumes because the radii differ. Both solids have a central hole in each interior cross-section, so subtract the inner disc.
Equivalent mathematically justified methods accepted.
Exact answers unless a decimal accuracy is specified.
Part a: Identify the outer and inner radii.
Part a: Set up the correct difference-of-squares integral.
Part a: Obtain 2 pi /15.
Part b: Rewrite the boundaries and set up the integral in y.
Part b: Obtain pi /6.
Part b: Obtain ratio 5/4.
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