why is an anti-derivative needed for integration
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Suppose you have a sequence of numbers, let us call it *s*, for example
>*s = 1, 3, 5, 7, 9,* that is *s_n=2n-1*
Now, let us calculate a sequence of sums of consecutive terms of the sequence *s,* that is
>*1*
>
>*1+3=4*
>
>*1+3+5=9*
>
>*1+3+5+7=16*
>
>*1+3+5+7+9=25*
We’ve obtained a new sequence, call it *S* that goes
>*S = 1, 4, 9, 16, 25,* that is *S_n=n^2*
Integration is basically this process of constructing *S* from *s*, just generalized so that it works with functions, not only sequences, but the idea is the same.
Can we get *s* back from *S*? Yes, easily.
The first term of *S* is the sum of the first one terms of *s*, so
>*S_1 = s_1*
Thus we deduce that the first term of *s* is *1.*
Next, for the second term of *S* we have
>*S_2 = s_1 + s_2 = S_1 + s_2*
>
>so *s_2=S_2 – S_1*
Thus *s_2=4-1=3.* Then
>S_3 = s_1 + s_2 +s_3 = (s_1 + s_2) + s_3 = S_2+s_3
>
>so *s_3=S_3 – S_2*
Thus *s_3=9-4=5.*
And so on, we find that the *n*th term of *s* is given by the difference *s_n=S_n – S_{n-1}*.
To get *s* from *S* we just needed to compute the differences between consecutive terms of *S.* We might say that we get *s* by differentiating *S.* Again, computing derivative is the same idea, only formalized in a way so that it works for functions. In this analogy,
>*S* is an integral of *s*
And since the derivative of *S* is *s,* we might say that
>*S* is an antiderivative of *s*
The fundamental theorem of calculus is then just a formal restatement of this observation.
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