$\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} \newcommand{\intd}[1]{\int\hspace{-0.7em}\int\limits_{\hspace{-0.7em}{#1}}} $

Chapter 4 Section 12: Leibniz' rule

Leibniz's rule
1. $\int_a^x f(t)dt = F(t)\Big|_a^x = F(x) - F(a)$
2. $ \frac{d}{dx}\int_a^x f(t)dt = f(x)$
3. $ \frac{d}{dx}\int_x^b f(t)dt = -f(x)$
4. $\fcolorbox{white}{yellow}{$ \frac{d}{dx}\int_{u(x)}^{v(x)} f(t)dt = f(v(x))\frac{dv}{dx} -f( u(x)) \frac{du}{dx} $}$
5. $\fcolorbox{white}{yellow}{$ \frac{d}{dx}\int_{u(x)}^{v(x)} f(x,t)dt = f(x,v(x))\frac{dv}{dx} -f(x, u(x)) \frac{du}{dx} + \int_{u(x)}^{v(x)} \frac{\p f}{\p x}(x,t)dt $}$

Chapter 5. Multiple Integrals

Double and triple integrals: Draw a picture and a small rectangle to determine the order of integration.
Applications
1. Arc Length:
  If $y = f(x)$, then
  
  $\fcolorbox{white}{yellow}{$ ds = \sqrt{ dx^2 + dy^2 } = \sqrt{1+ \Big(\frac{dy}{dx}\Big)^2} dx $}$
  ;
2. Center of mass $(\bar x, \bar y, \bar z)$:
  
  $\fcolorbox{white}{yellow}{$ \int \bar x dM = \int x dM, \m \int \bar y dM = \int y dM, \m \int \bar z dM = \int z dM, \m dM = \rho(x,y,z)dxdydz $}$
  ;
3. Moment of innertia with respect to coordinate axises for an area on $x$-$y$ plane:
  
  $\fcolorbox{white}{yellow}{$I_x = \int y^2 dM, \m I_y = \int x^2 dM, \m I_z = \int (x^2+y^2) dM $}$
  
  In general,
  $\fcolorbox{white}{yellow}{$\m I = \int (\text{distance})^2 dM$}$.
4. Volume by rotation: for solid volume generated by rotating $y=f(x)$ along the $x$-axis,
  Volume:
  $\fcolorbox{white}{yellow}{$ V = \int_a^b \pi \Big(f(x)\Big)^2 dx $}$
  , Surface area:
  $\fcolorbox{white}{yellow}{$ S= \int_a^b 2\pi f(x) ds = \int_a^b 2\pi f(x)\sqrt{1+(f'(x))^2}dx $}$
Change of variables; Jacobians
1. Polar coordinate:$\fcolorbox{white}{yellow}{ $x=r\cos\theta, \m y=r\sin\theta,$}$
  
  Area: $\fcolorbox{white}{yellow}{$ dA= rdrd\theta$}$.
  
  Arc length: $\fcolorbox{white}{yellow}{$r=r(\theta)$, or $\theta = \theta(r),$}$
  $\fcolorbox{white}{yellow}{$ds^2 = dr^2+r^2 d\theta^2, \mm $ $ds = \sqrt{\Big(\frac{dr}{d\theta}\Big)^2+ r^2} d\theta = \sqrt{1+r^2\Big(\frac{d\theta}{dr}\Big)^2}dr$}$
2. Cylindrical coordinate:$\fcolorbox{white}{yellow}{ $x=r\cos\theta, \m y=r\sin\theta, \m z= z$}$
  
  Volume : $\fcolorbox{white}{yellow}{$ dV= rdrd\theta dz$}$.
  $\mm$
  Surface area at $r=a$: $\m dA = a d\theta dz$
  
  Arc length: $\m ds^2 = dr^2+r^2 d\theta^2+dz^2, \mm $
3. Moment of innertia with respect to coordinate axises:
  $\mm I= \int (distance)^2 dM, \m dM = \rho dV$.
  e.g., Rotate about $z$-axis: $\m I = \int\hspace{-0.7em}\intd{} (x^2+y^2) \rho(x,y,z)dxdydz $

Surface integrals (formula will be provided)