Dror Bar-Natan: Classes: 2002-03: Math 157 - Analysis I:

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Welcome Back!

armed with the known, we sail to explore the yet unknown

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The Known:

Setting. $f$ bounded on $[a,b]$, $P: a=t_0<t_1<\dots<t_n=b$ a partition of $[a,b]$, $m_i=\inf_{[t_{i-1},t_i]}f(x)$, $M_i=\sup_{[t_{i-1},t_i]}f(x)$, $L(f,P)=\sum_{i=1}^n m_i(t_i-t_{i-1})$, $U(f,P)=\sum_{i=1}^n M_i(t_i-t_{i-1})$, $L(f)=\sup_P L(f,P)$, $U(f)=\inf_P U(f,P)$. Finally, if $U(f)=L(f)$ we say that ``$f$ is integrable on $[a,b]$'' and set $\int_a^b f=\int_a^b f(x)dx=U(f)=L(f)$.

Theorem 13-1. For any two partitions $P_{1,2}$, $L(f,P_1)\leq U(f,P_2)$.

Theorem 13-2. $f$ is integrable iff for every $\epsilon>0$ there is a partition $P$ such that $U(f,P)-L(f,P)<\epsilon$.

Theorem 13-3. If $f$ is continuous on $[a,b]$ then $f$ is integrable on $[a,b]$.

Theorem 14-2. (The Second Fundamental Theorem of Calculus) If $f'$ is integrable on $[a,b]$, where $f$ is some differentiable function, then $\int_a^b f' = f(b) - f(a)$.

Theorem 13-4. If $a<c<b$ then $\int_a^b f=\int_a^c f+\int_c^b f$ (in particular, the rhs makes sense iff the lhs does).

The Yet Unknown:

Convention. $\int_a^a f:=0$ and if $b<a$ we set $\int_a^b f := - \int_b^a f$.

Theorem 13-4'. $\int_a^b f=\int_a^c f+\int_c^b f$ so long as all integrals exist, no matter how $a$, $b$ and $c$ are ordered.

Theorem 13-5. If $f$ and $g$ are integrable on $[a,b]$ then so is $f+g$, and $\int_a^b f+g = \int_a^b f + \int_a^b g$.

Theorem 13-6. If $f$ is integrable on $[a,b]$ and $c$ is a constant, then $cf$ is integrable on $[a,b]$ and $\int_a^b cf = c\int_a^b f$.

Theorem 13-7^a. If $f\leq g$ on $[a,b]$ and both are integrable on $[a,b]$, then $\int_a^b f\leq\int_a^b g$.

Theorem 13-7. If $m\leq f(x)\leq M$ on $[a,b]$ and $f$ is integrable on $[a,b]$ then $m(b-a)\leq\int_a^b f\leq M(b-a)$.

Theorem 13-8. If $f$ is integrable on $[a,b]$ and $F$ is defined on $[a,b]$ by $F(x)=\int_a^x f$, then $F$ is continuous on $[a,b]$.

Theorem 14-1. (The First Fundamental Theorem of Calculus) Let $f$ be integrable on $[a,b]$, and define $F$ on $[a,b]$ by $F(x)=\int_a^x f$. If $f$ is continuous at $c\in[a,b]$, then $F$ is differentiable at $c$ and $F'(c)=f(c)$.

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