Double Integrals

Formal Definition

• Assume
  1. $D$ is a bounded closed region on a plane
  2. $f(x,y)$ is bounded on $D⟹m\le f(x,y)\le M\forall (x,y)\in D$
• Find
  • $R=[a,b]\times [c,d]$
  • $R_{kn}=[x_{k-1},x_k]\times [y_{k-1},y_k]$
    • $x_k=a+\Delta xk,k=0\dots N_1$
    • $x_n=c+\Delta yn,n=0\dots N_2$
  • $$\begin{gathered} m_{kn}={\inf }_{R_{kn}}f(x,y) \\ {M}_{kn}={\sup }_{R_{kn}}f(x,y) \end{gathered}$$
  • $$\begin{gathered} L(f,N_1,N_2)={\sum }_{k=1}^{N_1}{\sum }_{n=1}^{N_2}{m}_{kn}\Delta A \\ U(f,N_1,N_2)={\sum }_{k=1}^{N_1}{\sum }_{n=1}^{N_2}{M}_{kn}\Delta A \end{gathered}$$
• Definition
  • If ${\lim }_{(N_1,N_2)\to \infty }L(f,N_1,N_2)={\lim }_{(N_1,N_2)\to \infty }U(f,N_1,N_2)$
  • Then $f$ is Riemann integrable on $D$ and the limit is called the Riemann integral of $f$ over $D$
  • Notated: ${\iint }_{D}f(x,y)dA$

Integrability Conditions

1. $f$ is continuous on D (for the scope of this course)
2. $D$ is bounded by a smooth curve (can be piecewise like in the case of a rectangle)

Solving Strategies

Fubini’s Theorem

• If $f$ is continuous on $[a,b]\times [c,d]=R$
• Then ${\iint }_{D}f(x,y)dA={\int }_a^b{\int }_c^{d}f(x,y)dydx={\int }_c^d{\int }_a^{b}f(x,y)dxdy$
• Means you can swap the order of integration

Factorization

$${\iint }_{D}fdA={\int }_a^b{\int }_c^{d}g(x)h(y)dxdy={\int }_a^{b}g(x)dx{\int }_c^{d}h(y)dy$$

Over General Regions

• General Region = $D$ is simple relative to the direction $\vec{u}$ if any line $\mathcal{L}\parallel \vec{u}$ intersects $D$ along one line segment
• When $D$ is vertically simple
  • ${\iint }_{D}f(x,y)dA={\int }_a^b{\int }_{y_b(x)}^{y_t(x)}f(x,y)dydx$
• When $D$ is horizontally simple
  • ${\iint }_{D}f(x,y)dA={\int }_a^b{\int }_{y_b(x)}^{y_t(x)}f(x,y)dydx$
• When $D$ is simple both directions, one way might be a lot easier
• If $D$ is not simple either way, it can be cut into $D_1,D_2,\dots$ then added together after individual integration

Properties

Area of D

$$A(D)={\iint }_{D}dA$$

Volume Under $z=f(x,y)$

$$V={\iint }_{D}f(x,y)dA\text{ when }f(x,y)\ge 0$$

Integral Mean Value Theorem (IMVT)

• If $f$ is continuous and integrable on a closed bounded $D$
• $\exists (x^{\ast },y^{\ast })\in D\text{ s.t. }{\iint }_{D}f(x,y)dA=A(D)f(x^{\ast },y^{\ast })$
  • Where $A(D)$ is the area of the region $D$
• Also
  • For $m\le f(x,y)\le M$ on $D$
  • $mA(D)\le {\iint }_{D}f(x,y)dA\le MA(D)$

Independence of a Partition

• We can take the Riemann integral using any shaped partition
• Foundation of changing variables in integrals

Change of Variables

• Polar Change of Variables
• General Double Change of Variables

Functions

• $\mathcal{F}(S,F)$ denotes all functions $f$ where $f:S\to F$
• Definitions ($f,g\in \mathcal{F}(S,F)$)
  • $f$ and $g$ are equal if $f(s)=g(s),\forall s\in S$
  • $\mathcal{F}(S,F)$ is a vector space with $(f+g)(s)=f(s)+g(s)$ and $(cf)(s)=c[f(s)]$
• Polynomials
  • $f(x)=a_n{x}^n+a_{n-1}{x}^{n-1}+a_{1}x+a_0$, each $a_k\in F$ (degree $n$)
  • If $a_n=a_{n-1}=\cdots =a_0=0$ then $f$ is the zero polynomial (degree -1)
  • Two polynomials of degree $n$ are equal if all coefficients are equal
  • The set of all polynomials of field F and degree $n$ form a vector space $P_n(F)$
    • Let $f(x)=a_n{x}^n+\cdots +a_{1}x+a_0$, $g(x)=b_m{x}^m+\cdots +b_{1}x+b_0$ where $m\le n$
    • Define $b_{m+1}=b_{m+2}=\cdots b_n=0$ (make $g$ degree n by padding with zero coefficients)
    • Define
      • $f(x)+g(x)=(a_n+b_n)x^n+(a_{n-1}+b_{n-1})x^{n-1}+\cdots +(a_1+b_1)x+a_0+b_0$
      • For any $c\in F$, $cf(x)=c{a}_n{x}^n+\cdots +c{a}_{1}x+c{a}_0$