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In my textbook, there are two fundamental theorems of Calculus. The first states:

Let $ f $ be integrable on $[a,b]$. For $x\in[a,b]$ put $F(x)=\int^x_a f(t)dt$. Then $ F$ is continuous on $[a,b]$ and furthermore if $ f $ is continuous at a point $x_0$ of $[a,b]$ then $F$ is differentiable at $x_0$ and $F'(x_0)=f(x_0)$.

And the second states:

If $ f $ is integrable on $[a,b]$ and if there is a differentiable function $ F $ on $[a,b]$ such that $F'=f$, then $\int^b_a f(x)dx=F(b)-F(a)$.

My question is; what is the difference between these two? when is each useful?

user444444242
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1 Answers1

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The first one tells you how to evaluate derivatives of functions defined by inverses ("the derivative of the integral is the function itself").

The second part tells you how to evaluate an integral using antiderivatives ("the integral of the antiderivative is the function itself").

So they're two sides of the same coin. Taken together they (informally) say "integration and differentiation are inverses of each other", separately they say "differentiation is the inverse of integration" and "integration is the inverse of differentiation".

Both are very useful, but the second part will be used more frequently in a calculus class. Recall that the definition of an integral is in terms of Riemann sums; it would be grueling to have to evaluate every integral using limits of sums. The fundamental theorem of calculus (part 2) gives you an (often easier) alternative: finding an antiderivative.

BaronVT
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