Fundamental Theorem of Calculus/Second Part/Proof 2

Theorem

Let $f$ be a real function which is continuous on the closed interval $\closedint a b$.

Then:

$(1): \quad f$ has a primitive on $\closedint a b$

$(2): \quad$ If $F$ is any primitive of $f$ on $\closedint a b$, then:

$\ds \int_a^b \map f t \rd t = \map F b - \map F a = \bigintlimits {\map F t} a b$

Proof

As $f$ is continuous, by the first part of the theorem, it has a primitive. Call it $F$.

$\closedint a b$ can be divided into any number of closed subintervals of the form $\closedint {x_{k - 1} } {x_k}$ where:

$a = x_0 < x_1 \cdots < x_{k-1} < x_k = b$

Fix such a finite subdivision of the interval $\closedint a b$; call it $P$.

Next, we observe the following telescoping sum identity:

\(\text {(1)}: \quad\)

\(\ds \sum_{i \mathop = 1}^k \paren {\map F {x_i} - \map F {x_{i - 1} } }\)

\(=\)

\(\ds \map F b - \map F a\)

Because $F' = f$, $F$ is differentiable.

By Differentiable Function is Continuous, $F$ is also continuous.

Therefore we can apply the Mean Value Theorem on $F$.

It follows that in every closed subinterval $I_i = \closedint {x_{i - 1} } {x_i}$ there is some $c_i$ such that:

$\map {F'} {c_i} = \dfrac {\map F {x_i} - \map F {x_{i - 1} } } {x_i - x_{i - 1} }$

It follows that:

\(\ds \map F {x_i} - \map F {x_{i - 1} }\)

\(=\)

\(\ds \map {F'} {c_i} \paren {x_i - x_{i - 1} }\)

\(\text {(2)}: \quad\)

\(\ds \map F b - \map F z\)

\(=\)

\(\ds \sum_{i \mathop = 1}^k \map {F'} {c_i} \paren {x_i - x_{i - 1} }\)

from $(1)$

\(\ds \)

\(=\)

\(\ds \sum_{i \mathop = 1}^k \map f {c_i} \paren {x_i - x_{i - 1} }\)

because $F' = f$

From the definitions of supremum and infimum, we have for all $i$ (recall $I_i = \closedint {x_{i - 1} } {x_i}$):

$\displaystyle \inf_{x \mathop \in I_i} \map f x \le \map f {c_i} \le \sup_{x \mathop \in I_i} \map f x$

From the definitions of upper and lower sums, we conclude for any finite subdivision $P$:

$\displaystyle \map L P \le \sum_{i \mathop = 1}^k \map f {c_i} \paren {x_i - x_{i - 1} } \le \map U P$

Lastly, from the definition of a definite integral and from $(2)$, we conclude:

$\displaystyle \map F b - \map F a = \int_a^b \map f t \rd t$

$\blacksquare$

Sources

• 2005: Roland E. Larson, Robert P. Hostetler and Bruce H. Edwards: Calculus (8th ed.): $\S 4.4$