$\newcommand{\dis}{\displaystyle} \newcommand{\m}{\hspace{1em}} \newcommand{\mm}{\hspace{2em}} \newcommand{\x}{\vspace*{1ex}} \newcommand{\xx}{\vspace*{2ex}} \let\limm\lim \renewcommand{\lim}{\dis\limm} \let\fracc\frac \renewcommand{\frac}{\dis\fracc} \let\summ\sum \renewcommand{\sum}{\dis\summ} \let\intt\int \renewcommand{\int}{\dis\intt} \newcommand{\sech}{\text{sech}} \newcommand{\csch}{\text{csch}} \newcommand{\Ln}{\text{Ln}} \newcommand{\p}{\partial} $

Lecture 19, 10/5/2022. This page is for Section 1 only.
ACMS 20550: Applied Mathematics Method I
Instructor: Bei Hu, b1hu@nd.edu, Hurley 174A

Chapter 5. Multiple integrals.

  1. Review of some integration formulas.
    1. Substitution: For functions like $\exp\;$
      $\mm\rule{0ex}{2ex}$
      , $\ln\;$
      $\mm\rule{0ex}{2ex}$
      , $\sin\;$
      $\mm\rule{0ex}{2ex}$
      , $\cos\;$
      $\mm\rule{0ex}{2ex}$
      , $\cdots$;
      use $u= \;$
      $\mm\rule{0ex}{2ex}$
      ,
    2. Integration by parts: Try to differentiate the following after integration by parts,
      (a) $\ln\;$
      $\mm\rule{0ex}{2ex}$
      , (b) polynomials, (c) $\cdots$;
    3. Partial fraction: decompose into
      (a) $\frac{A}{x+a}+\frac{B}{x+b}+\cdots$, where $a\neq b$; $\cdots$,
      (b) $\frac{Ax+B}{(x+a)^2+b^2}+\cdots$,
      (c) $\cdots$;
  2. Double and tripple integrals.
    1. Draw a picture;
    2. Determine the order of integration;
    3. Determine the upper and lower limit.