why is an anti-derivative needed for integration
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The derivative is the opposite of the integral and vice versa. The anti-derivative(integral) is needed for integration because integration is the act of taking the integral of a function.
The derivative of a function its its change over time.
The integral changes based on the value of the function. If you’re integrating something with the value of 1, then at that point the integral will have a slope of 1.
In this way the integral is *almost* the opposite of the derivative. There is something missing, though. Take f(x)=1 as an example. The derivative is 0. Taking the anti-derivative of f(x)=0 gives us f(x)=0 again, so we aren’t actually back where we started until we add a constant.
The anti-derivative isn’t necessarily *needed* for integration. You could theoretically solve an integral by summing up the area under the curve through brute-force methods like Riemann sums.
However, the entire point of calculus is that derivatives and integrals are the opposite of each other. It’s such a key point that it’s literally named the “Fundamental Theorem of Calculus”.
So if you can do a derivative, and you can figure out how to do it backwards, then you can do an integral. It’s not that it’s *needed*, it’s that it’s a way easier method than trying to brute-force it.
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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