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[parent] proof of the fundamental theorem of calculus (Proof)

Recall that a continuous function is Riemann integrable, so the integral

$\displaystyle F(x) = \int_c^x f(t)\, dt $
is well defined.

Consider the increment of $ F$:

$\displaystyle F(x+h)-F(x) = \int_c^{x+h} f(t)\, dt - \int_c^x f(t)\, dt = \int_x^{x+h} f(t)\, dt $
(we have used the linearity of the integral with respect to the function and the additivity with respect to the domain).

Now let $ M$ be the maximum of $ f$ on $ [x,x+h]$ and $ m$ be the minimum. Clearly we have

$\displaystyle m h \le \int_x^{x+h} f(t)\, dt \le M h $
(this is due to the monotonicity of the integral with respect to the integrand) which can be written as
$\displaystyle \frac{F(x+h)-F(x)}{h} = \frac{\int_{x}^{x+h}f(t)\, dt}{h} \in [m,M] $

Since $ f$ is continuous, by the mean-value theorem, there exists $ \xi_h\in [x,x+h]$ such that $ f(\xi_h) = \frac{F(x+h)-F(x)}{h}$ so that

$\displaystyle F'(x)= \lim_{h\to 0} \frac{ F(x+h)-F(x)}{h} = \lim_{h\to 0} f(\xi_h) = f(x) $
since $ \xi_h\to x$ as $ h\to 0$. This proves the first part of the theorem.

For the second part suppose that $ G$ is any antiderivative of $ f$, i.e. $ G'=f$. Let $ F$ be the integral function

$\displaystyle F(x)= \int_a^x f(t) \, dt. $
We have just proven that $ F'=f$. So $ F'(x)=G'(x)$ for all $ x\in [a,b]$ or, which is the same, $ (G-F)'=0$. This means that $ G-F$ is constant on $ [a,b]$ that is, there exists $ k$ such that $ G(x)=F(x)+k$. Since $ F(a)=0$ we have $ G(a)=k$ and hence $ G(x)=F(x)+G(a)$ for all $ x\in[a,b]$. Thus
$\displaystyle \int_a^b f(t)\, dt = F(b) = G(b) - G(a). $



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Cross-references: antiderivative, mean-value theorem, monotonicity, domain, additivity, function, well defined, integral, Riemann integrable, continuous function

This is version 6 of proof of the fundamental theorem of calculus, born on 2003-07-17, modified 2006-09-01.
Object id is 4463, canonical name is ProofOfTheSecondFundamentalTheoremOfCalculus.
Accessed 10914 times total.

Classification:
AMS MSC: 26-00 (Real functions :: General reference works )
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