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Q&A #17201 |
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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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