I always thought integrals and anti derivatives were the same thing but I recently read that they are separate things that can be related by the Fundamental Theorem of Calculus. How does that work? Tried reading a couple explanations but they didn’t dumb it down enough for me I think.
In: Mathematics
Suppose you start with a sequence S that has terms s_1, s_2, s_3, … and so on.
Now let’s introduce an operation D that will assign to a sequence S a new sequence D(S) whose terms are the differences s_2-s_1, s_3-s_2, … and so on. For example, if S=(2,4,6,8,10,…), then D(S)=(2,2,2,2,…), or if S=(1,4,9,16,…), then D(S)=(3,5,7,…).
Now let’s introduce another operation P that will assign to a sequence S a new sequence P(S) whose terms are partial sums s_1, s_1+s_2, s_1+s_2+s_3, … and so on. For example, if S=(1,1,1,1,…), then P(S)=(1,2,3,4,…), or if S=(1,10,100,1000,…), then P(S)=(1,11,111,1111,…).
Those two operations seem unrelated at the first glance. However, let’s for the sake of illustration consider a sequence S=(0,1,4,6,8,9,…). Then the sequence D(S) will be (1,3,2,2,1,…). Now, D(S) is a sequence too, so we can perform operation P on it. We obtain P(D(S))=(1,4,6,8,9,…). That looks very similar to the original sequence S, except we skipped the first term.
Indeed, looking at what happens closely, the first few terms of sequence D(S) were (1-0,4-1,6-4,…) and then for P(D(S)) were (1-0,(1-0)+(4-1),(1-0)+(4-1)+(6-4),…) and you can see the pattern. As (1-0)+(4-1)=(4-1)+(1-0)=4-1+1-0=4+(-1+1)-0=4-0=4 and (1-0)+(4-1)+(6-4)=(6-4)+(4-1)+(1-0)=6-4+4-1+1-0=6+(-4+4)+(-1+1)-0=6-0=6, we see how we recovered the terms of the original sequence apart from the first that was 0.
It looks like the operation P is kind of opposite to operation D. Here’s where people who know calculus should be able to tell where I’ve tricked you somehow.
I purposefuly used a sequence S that started with a zero. Notice, for example that if I started with a sequence T=(2,3,6,8,10,11,…) which is just sequence S+2=(0+2,1+2,4+2,6+2,8+2,9+2,…) (I added the constant 2 to every term of S), the sequence D(T)=(1,3,2,2,1,…) is the same as D(S) and consequently P(D(T)) would be the same as P(D(S)) as well. So the composite operation PD doesn’t yield the original sequence, only the sequence shifted by a constant.
Now doing all this rigorously and for continuous functions instead of sequences, the D would be the differentiation operation, the P would be integration operation and the relationship that PD returns the original function up to shift by a constant is the main idea of the fundamental theorem of calculus that says that under certain conditions you can calculate the integral (a kind of a sum) with the help of antiderivative (formally reversing the differentiation operation).
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