>Consider the region bounded above by y = 1/x. Compute the area under
>this curve, above the x-axis, between x = 1 and x = 00 (infinity).
>The "area" of this region is given by fnInt(1/x,x,1,00) and is
>infinite (the improper integral diverges).
>
>However the volume obtained when we rotate this region about the x-axis
>is given by pi*fnInt(1/x*x,x,1,00) = pi.
>
>How do you expain to students that the area is infinite but the volume
>is finite, since the area fits inside the volume?
>
>Brian
and I would like to share some obvious thoughts on it.
First, what does it MEAN to explain something to someone? I would understand
an 'explanation' to be an attempt (hopefully successful, though frequently
not, and so make another attempt) at giving some insight to someone who has
thought about something that has perplexed them. Brian Smith's question does
raise a point that will surely perplex students (well, weak ones certainly).
I would like to suggest that the ROOT cause of difficulty in connection with
Brian Smith's question is a (possible) lack of understanding on behalf of
the learner in connection with the behaviour of the following INFINITE sums,
the first of which is 'divergent' (by which it MUST be understood that the
'partial sums' can be made 'arbitrarily large', which in plain language just
means that by adding together ENOUGH of them one may produce a number that
is greater than ANY pre-chosen number), and the second of which is
'convergent' (by which it must be understood that although the 'partial
sums' get greater and greater the more terms are added together,
nevertheless the partial sums can NOT be made arbitrarily large; the partial
sums are BOUNDED):
(1/1 + 1/2 + 1/3 + 1/4 + ... ) (related to the area)
and
(1/1 1/2 1/3 1/4 ... ) (related to the volume)
I realise that this in an imperfect suggestion - it is not possible to trash
out all the difficulties in a brief e-mail message - but it is my belief
that UNLESS a student had that deeply ingrained numerical understanding then
he/she would have NO real hope of 'understanding' WHY the area is infinite
and the volume is finite.
Let's face it, trying to explain infinite processes is what makes teaching
Calculus/Analysis the challenge that it is.
Best wishes to you all,
John Cosgrave,
Mathematics Department,
St. Patrick's College,
Drumcondra,
Dublin 9,
IRELAND.
P.S. I don't have an e-mail address at my College (the Administration ... ).
The above is my private e-mail address.