Dror Bar-Natan: Classes: 2002-03: Math 157 - Analysis I: | (147) |
Next: Homework Assignment 14
Previous: Class Notes for the Week of December 2 (8 of 8) |

this document in PDF: WelcomeBack.pdf

**The Known:**

**Setting. ** bounded on ,
a partition of ,
,
,
,
,
,
. Finally, if we say
that `` is integrable on '' and set
.

**Theorem 13-1. ** For any two partitions ,
.

**Theorem 13-2. ** is integrable iff for every
there is a partition such that
.

**Theorem 13-3. ** If is continuous on then is
integrable on .

**Theorem 14-2. ** (The Second Fundamental Theorem of
Calculus) If is integrable on , where is some
differentiable function, then
.

**Theorem 13-4. ** If then
(in particular, the rhs makes sense iff
the lhs does).

**The Yet Unknown:**

**Convention.**
and if we set
.

**Theorem 13-4'. **
so
long as all integrals exist, no matter how , and are ordered.

**Theorem 13-5. ** If and are integrable on
then so is , and
.

**Theorem 13-6. ** If is integrable on and is a
constant, then is integrable on and
.

**Theorem 13-7 ^{a}. **
If on and both are integrable on , then
.

**Theorem 13-7. ** If
on and is
integrable on then
.

**Theorem 13-8. ** If is integrable on and is
defined on by
, then is continuous on .

**Theorem 14-1. ** (The First Fundamental Theorem of
Calculus) Let be integrable on , and define on by
. If is continuous at , then is
differentiable at and
.

The generation of this document was assisted by
L^{A}TEX2`HTML`.

Dror Bar-Natan 2003-01-07