**Problem**

**Introduction**

This is a
problem involving the calculation of the volume of solid of revolution of an
enclosed region. The junior colleges
(or senior high schools) like to set this type of question.

**Technique**

If the axis
of revolution is the

*y*-axis, the basic formula is ò*p**x*² d*y*with the appropriate lower and upper limits. Notice that this only works for the region between one curve and the axis and when rotated, this will generate a solid with no hollow parts. An enclosed region, however, consists of two curves. In our case, when we make*x*the subject, we find that we have two choices. One of them leads to a curve that is further away from the axis of rotation. I call that the*outer*curve. The other curve is the*inner*curve, and this is nearer the axis of rotation. We need to subtract the volume generated by the inner curve from that generated by the outer curve.**Solution**

**Remarks**

In this
example, the curve on the right happens to be the outer curve. If the equation were

(

*x*+ 93)² +*y*² = 15², the circular region would be on the left of the*y*-axis and the outer curve would be on the left.
For your information, the above solid of revolution is
a

*torus*. This is the shape of a doughnut, (or hoopla-hoop, circular tube, or Polo mint perhaps?). It is the inner curve that gives the hole in the “doughnut”.
You can imagine
in your mind’s eye that as the circular disk revolves around the

*y*-axis, its centre traces out a circular path of 93 units. By the Second Centroid Theorem of Pappus,
volume = length of path of centroid × area of
cross section = 2

*p*(93) ×*p*(15)² = 41850*p*²
In general, the volume of a torus with major
radius

*R*and minor radius*r*is
volume = 2

*p**R*×*p**r*² = 2*Rr*²*p*²
If you know this fact, you use it to check your calculations. Although this is not in the H2 Syllabus, but it
is something interesting to explore.

H02. Use a
diagram / model

H04. Look for
pattern(s)

H05. Work
backwards
[e.g. making

*x*the subject]
H09. Restate
the problem in another way [symmetry:
volume is twice of upper half]

H10. Simplify
the problem
[integration by substitution]

H11. Solve part
of the problem

H13* Use
Equation / write a Mathematical Sentence

**Suitable Levels**

*****GCE ‘A’ Levels H2 Mathematics

* IB Mathematics HL (Applications of
Integration)

* Advanced Placement (AP) Calculus AB
& BC

* University / College Calculus

* other syllabuses that involve Applications of Integration

* any precocious or independent learner who
loves to learn

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