Does the Fundamental Theorem of Calculus tell us that integration is the ‘opposite’ of differentiation

calculusdefinite integralsderivativesindefinite-integralsintegration

I have often read that the Fundamental Theorem of Calculus (FTC) tells us that integration is the opposite of differentiation. I have always found this summary confusing, so I will lay out what I think people mean when they make such a statement.

The First FTC implies the existence of antiderivatives for every function, $f$, that is continuous on a particular interval, say $[a,b]$. Generally, we denote this antiderivative as $F$. Differentiating $F$ gets back to our original function, $f$. So when people say that 'integration is the opposite of differentiation', what they mean is that an antiderivative of a function can be computed using a definite integral.

The Second FTC is more powerful than the First FTC, as it tells us that definite integrals can be computed using the antiderivative of a function (which is generally more useful than knowing that one possible antiderivative of $f$ can be computed using a definite integral, $F$). For the Second FTC, I don't understand how this is related to 'integration being the opposite of differentiation' at all. The Second FTC shows us the link between antiderivatives (indefinite integrals) and definite integrals. It is extremely useful for trying to find the area under a curve, but I'm not sure how this relates to integration and differentiation being 'opposites'.

Is there something about the First FTC or the Second FTC that has a bigger implication about integration being the opposite of differentiation, or is my understanding correct?

Best Answer

I think the first FTC:

If $f: [a,b] \to \Bbb R$ is continuous then $F: [a,b] \to \Bbb R$ defined by $F(x)=\int_a^x f(t)dt$ is differentiable and $F'(x)=f(x)$ for all $x \in [a,b]$.

is what people mean by saying the integration (which defines $F$) is the inverse of differentiation (as we have found a function with derivative $f$).

The second FTC

If $f: [a,b] \to \Bbb R$ is Riemann-integrable on $[a,b]$ and we have a function $F: [a,b] \to \Bbb R$ such that $F'(x)=f(x)$ on $[a,b]$, then $\int_a^b f(x)dx=F(b)-F(a)$.

is more of a "recipe" to find an integral: the target is to compute the definite integral and the tool we're given is to find an antiderivative. So not an inverse as such but a method. It's a bit of an iffy one, as an antiderivative $F$ need not exist at all (except when $f$ is continuous and the first FTC gives us one, but not explicitly, but at least we know some solution exists, but we don't have it in computable form yet). I think the first is closer to giving a direct "inverse" connection between integration and differentiation (and is often used in other contexts when we differentiate wrt boundaries of integrals, etc.). But that's just one view.

The first FTC can be summarised as $$\frac{d}{dx}\int_a^x f(t)dt = f(x)$$ so "Applying the integration operator to $f$, followed by the differentiation operator gives us back $f$ again".

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