Dror Bar-Natan: Classes: 2002-03: Math 157 - Analysis I: (137) Next: Homework Assignment 12a
Previous: Solution of Term Exam 2

Integration

this document in PDF: Integration.pdf

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

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

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

Theorem 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).

Theorem 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 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 $ \mathbf{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 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 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]$.



This class' fundamental existential dilemma / schizophrenia:

The generation of this document was assisted by LATEX2HTML.



Dror Bar-Natan 2003-01-06