Question 8.T.12: (Fundamental Theorem of Calculus) If F is differentiable on ...
(Fundamental Theorem of Calculus)
If F is differentiable on [a, b] and F^{\prime} ∈ \mathcal{R}(a, b) , then
\int_{a}^{b}{F^{\prime} (x) dx} = F (b) − F (a) . (8.17)
If P = \left\{x_{0}, x_{1}, …, x_{n}\right\} is any partition of [a, b] , then
F (b) − F (a) = \sum\limits_{i=0}^{n−1}{[F (x_{i+1}) − F (x_{i})]} .
Applying the mean value theorem to F on [x_{i}, x_{i+1}] , we can choose α_{i} ∈ (x_{i}, x_{i+1}) such that
F (x_{i+1}) − F (x_{i}) = F^{\prime}(α_{i}) (x_{i+1} − x_{i}) .
Consequently
F (b) − F (a) = \sum\limits_{i=0}^{n−1}{F^{\prime}(α_{i}) (x_{i+1} − x_{i})} = S \left(F^{\prime}, P, α \right) ,
where α = (α_{0}, α_{1}, . . . , α_{n−1}) is a mark on P.
Now choose a sequence (P_{k}) of partitions such that \left\|P_{k}\right\| → 0, and a corresponding sequence of marks(α_{k}) determined by the mean value theorem as above. We then have
F (b) − F (a) = S \left(F^{\prime}, P_{k}, α_{k} \right) , k ∈ \mathbb{N}.
Since F^{\prime} ∈ \mathcal{R}(a, b) , the right hand side converges to \int_{a}^{b}{F^{\prime} (x) dx} as k → ∞.