121616949-math.158 - 144 Chapter 7 Integration time It is...

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144 Chapter 7 Integration time. It is certainly true that it is somewhere, so let’s suppose that at t = 0 the position is k . Then just as in the example, we know that the position of the object at any time is 3 t 2 / 2+ k . This means that at time t = a the position is 3 a 2 / 2 + k and at time t = b the position is 3 b 2 / 2 + k . Therefore the change in position is 3 b 2 / 2 + k - (3 a 2 / 2 + k ) = 3 b 2 / 2 - 3 a 2 / 2. Notice that the k drops out; this means that it doesn’t matter that we don’t know k , it doesn’t even matter if we use the wrong k , we get the correct answer. In other words, to find the change in position between time a and time b we can use any antiderivative of the speed function 3 t —it need not be the one antiderivative that actually gives the location of the object. What about the second approach to this problem, in the new form? We now want to approximate the change in position between time a and time b . We take the interval of time between a and b , divide it into n
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