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
An antiderivative of a function f(x) is simply another function F(x) such that taking the derivative of F’(x) = f(x).
For example, consider f(x) = x. An antiderivative F(x) could be x^2 / 2 because F’(x) = 2x / 2 = x = f(x). But observe that this isn’t the ***only*** antiderivative, because say G(x) = x^2 / 2 + 7 ***also*** has the derivative G’(x) = 2x / 2 = x = f(x). This is because the derivative of any constant is 0. You may know this as adding “+ C” to the end of your antiderivative.
What the Fundamental Theorem of Calculus says is that ***if*** you are trying to find the integral from a to b of f(x) ***and if*** you can find an antiderivative F(x), then the integral = F(b) – F(a), whatever antiderivative you pick.
So again, let’s go back to f(x) = x and integrate from 0 to 4. If you draw this on a graph, observe it’s a triangle. It has corners (0,0); (4,4); and (4,0). And we ***know*** the area of a triangle, so we know this should have area = 1/2 x base x height = 1/2 x 4 x 4 = 8.
Now let’s take our antiderivative F(x) = x^2 / 2 from earlier. F(4) = 4^2 / 2 = 16/2 = 8. F(0) = 0. Thus, it is ***true*** that F(4) – F(0) = 8 – 0 = 8 = the integral.
But we can ***also*** use our other antiderivative G(x). Observe G(4) = 4^2 / 2 + 7 = 8 + 7 = 15 and G(0) = 7. Thus G(4) – G(0) = 15 – 7 = 8 still.
At a high-level, the two things are answering different questions. An antiderivative answers “what function exists that has the derivative of a given function.” An integral answers “what is the signed area under the curve.” In most cases, you’ll need an antiderivative in order to calculate an integral.
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