George,
I'm afraid I do not really understand what you are saying.
The formula for the length of arc of a function f(x) from 0 to t, call it
s(t), is given by the definite integral from 0 to t of sqrt(1+(f ' (x))^2 ).
DERIVE has no problem with the little program :
#1 f(x):= , #2 s(t):=int(sqrt(1+(f ' (x))^2 ),x,0,t).
If, for example, you want the length of arc of the parabola y=(1/2)x^2 from 0
to t, type in #3 f(x):= (1/2)x^2 and simplify (selecting BASIC) the above #2,
to obtain
(1/2)ln (sqrt(t^2+t)+(1/2)tsqrt(t^2+1). And if you want the length of arc for
this parabola from 0 to 2, then simplify s(2) (try out "equals"(=) and
"approximately equals"). Of course if you want the length of arc from a to b,
evaluate s(b)-s(a). Students should be able to find that out for themselves.
When you evaluated the int(sqrt(1+u^2),u,0,t) you actually determined a
formula for the length of arc of the parabola y=(1/2) from 0 to t. (By the
the way, I assume you intended (1+u^2)^(1/2) instead of (2+u^2)^(1/2)) .) I
do not know whether this is any help. Perhaps we are talking at cross
purposes.
Cheers,
Wim de Jong
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