The Fundamental Theorem of Calculus

Section 7.3 The Fundamental Theorem of Calculus

For good measure, let’s state the Big Theorem that we all know and love about the definite integral:

Theorem 7.3.1. Fundamental Theorem of Calculus.

If \(f:[a,b]\to\mathbb{R}\) is continuous, then the function \(\displaystyle F(x)=\int_{[a,x]}f(t)\ dt\) is differentiable. Moreover, for each \(x\in [a,b]\text{,}\) \(F'(x)=f(x)\text{.}\)

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But wait, that isn’t the theorem we all know and love. The one we all know and love is this one:

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Theorem 7.3.2. The Other Fundamental Theorem of Calculus.

If \(f:[a,b]\to \mathbb{R}\) is differentiable, and \(f'\) is integrable, then \(\displaystyle\int_{[a,b]}f'(x)\ dx=f(b)-f(a)\text{.}\)

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Proof.

For any partition \(P=\left\{a,x_1,\ldots,x_{n-1},b\right\}\text{,}\) each subinterval \([x_k,x_{k+1}]\) has some \(c_k\) so that \(f'(c_k) (x_{k+1}-x_k)=f(x_{k+1})-f(x_k)\text{.}\) So for each partition \(P\text{,}\)

\begin{equation*} \displaystyle L(f';P)\leq \sum_{k=0}^{n-1}f(x_{k+1})-f(x_k)\leq U(f';P)\ \ \ . \end{equation*}

Now, regardless of the partition, \(\displaystyle \sum_{k=0}^{n-1}f(x_{k+1})-f(x_k)=f(b)-f(a)\text{.}\)

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