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Q&A #17201


Integration

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From: Pat Ballew (for Teacher2Teacher Service)
Date: Jun 01, 2006 at 13:49:15
Subject: Re: Integration

The symbol for integration is the script sigma, for sum, and integration is
essentially addition.

Suppose a car increases velocity from dead still to 30 feet per second at a
steady rate over five seconds so that his velocity = 6s where s is the
number of seconds he has been moving, how far did the care go? We could get a
rough approximation by finding out how fast he was going end of each second,
and then assuming that the speed was (about) the same for the whole second,
i.e., from 0 to 1 or 1 to 2, etc. At 1 second he would be going 6 feet per
second, so from time 0 to time 1 he would travel approx. 6 feet, at 2 seconds
he would be going 12 feet per second, so from time 1 to time 2 he would
travel approx. 12 feet (12 feet per second for one second).

IF you did this for all five second intervals you would get a total distance
traveled of 6 + 12 + 18 + 24 + 30 or 90 feet for the total distance. But you
would realize that you overestimated his speed for most of the intervals, so
this is probably a little high.

Of course, if we wanted to be more accurate, we would calculate the speed
every 1/2 second, or every quarter, or every tenth of a second.  If we did
that (check my work) you would get distances of 82.5 feet, and then 78.75.
Each getting closer to the truth (but none of them exactly right).

What integration does is calculate the limit of this pattern of numbers as
we make the number of times we check get very large (goes to infinity). So
the integral is just the limit of the sums of the approximations, and
amazingly, it gives the EXACT answer. Gotta love math. Way to go Newton!

 -Pat Ballew, for the T2T service

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