Integrals

Rectangular Domains

Definition: Suppose $f:U\subset {\mathbb{R}}^2\to \mathbb{R}$ and $U$ is some rectangular region $[a,b]\times [c,d]$ ($\times$ is called the Cartesian product). The volume enclosed by the domain $U$ and the graph of $f$ is found with the double integral of $f$ over $U$, denoted by ${\iint }_{U}f(x,y)dA$ where $dA$ symbolically means $dxdy$.

Example: $f:[a,b]\times [c,d]\to \mathbb{R}$, $f(x,y)=k$
${\iint }_{[a,b]\times [c,d]}f(x,y)dxdy=k|a-b||c-d|$, this is a box

Example: $f:[0,1]\times [0,1]\to \mathbb{R}$, $f(x,y)=1-x$
${\iint }_{[0,1]\times [0,1]}f(x,y)dxdy=\frac{1\cdot 1\cdot 1}{2}=\frac{1}{2}$, this is a triangular prism

Needing to visualize the graphs in order to find the area is not ideal.

${\int }_{a_1}^{a_2}f(x)dx\approx f(a_1)(a_2-a_1)$ is the basis of Riemann sums

For a given partition $\left\{a_1,a_2,\dots ,a_n\right\}$
If ${\lim }_{n\to \infty }{\sum }_{i=1}^{n}f(a_i)(a_{i+1}-a_i)$ exists, we call the limit a Riemann integral

Theorem: ${\lim }_{n\to \infty }\sum f(a_i)(a_{i+1}-a_i)$ exists for $f:[a,b]\to \mathbb{R}$ when $f$ is almost everywhere continuous

Definition: Cavalieri’s principle states that the volume of a body can be given by ${\int }_a^{b}A(x)dx$ where $a$ and $b$ are minimum and maximum distances from the reference plane and $A(x)$ denotes the cross-sectional area of the volume measured at a distance $x$ from a reference plane. This is also called the slice method.

This helps lead to the idea that volume can be represented with an iterated integral ${\int }_a^b{\int }_c^{d}f(x,y)dydx$

To calculate an iterated integral $\iint f(x,y)dxdy$ analytically, find it with regards to $x$ and then $y$ (or vice versa)

Theorem: Fubini’s theorem says that with a function $f$ continuous (or where the discontinuities lie on a finite union of graphs of continuous functions) on the rectangular domain $[a,b]\times [c,d]$, ${\iint }_{R}f(x,y)dA={\int }_a^b\left({\int }_c^{d}f(x,y)dy\right)dx={\int }_c^d\left({\int }_a^{b}f(x,y)dx\right)dy$

Let’s define a more rigorous definition of the integral in terms of Reimann sums. Consider a closed rectangle $R\subset {\mathbb{R}}^2=[a,b]\times [c,d]$. A regular partition of $R$ in order $n$ means two ordered collections of $n+1$ equally spaced points ${\left\{x_j\right\}}_{j=0}^n$ and ${\left\{y_k\right\}}_{k=0}^n$ which divide up $R$ into rectangles. It follows that $x_{j+1}-x_j=\frac{b-a}{n}$ and $y_{k+1}-y_k=\frac{d-c}{n}$

A function $f(x,y)$ is bounded if there is a number $M>0$ such that $-M\le f(x,y)\le M$ for all $(x,y)$ in the domain of $f$. A continuous function on a closed rectangle is always bounded.

Definition: Let $R_{jk}$ be $[x_j,x_{j+1}]\times [y_k,y_{k+1}]$ and let ${\mathbf{c}}_{jk}$ be any point in $R_{jk}$. Suppose $f:R\to \mathbb{R}$ is a bounded real-valued function. Form the sum $S_n={\sum }_{j,k=0}^{n-1}f({\mathbf{c}}_{\mathbf{jk}})\Delta x\Delta y$ or $\Delta A=\Delta x\Delta y$ where $\Delta x=x_{j+1}-x_j$ and $\Delta y=y_{k+1}-y_k$. Essentially, this Reimann sum adds up $n^2$ rectangles.

Definition: If $\left\{S_n\right\}$ converges to a limit $S$ as $n\to \infty$ and $S$ is the same for any choice of points ${\mathbf{c}}_{jk}$ then $f$ is integrable over $R$ and we write ${\iint }_{R}f(x,y)dA={\iint }_{R}f(x,y)dxdy={\iint }_{R}fdxdy$

${\lim }_{n\to \infty }{\sum }_{j,k=0}^{n-1}f({\mathbf{c}}_{jk})\Delta x\Delta y={\iint }_{R}fdxdy$ for any choice ${\mathbf{c}}_{jk}\in R_{jk}$

Theorem: Any continuous function defined on a closed rectangle $R$ is integrable

Theorem: Let $f:R\to \mathbb{R}$ be a bounded real-valued function on the rectangle $R$ and suppose that the set of points where $f$ is discontinuous lies on a finite union of graphs of continuous functions. Then $f$ is integrable over $R$.

Some properties follow easily

1. ${\iint }_R(f+g)dA={\iint }_{R}fdA+{\iint }_{R}gdA$
2. ${\iint }_{R}cfdA=c{\iint }_{R}fdA$
3. $f\ge g⟹{\iint }_{R}fdA\ge {\iint }_{R}gdA$
4. If $Q=R_1\cup R_2\cup \cdots \cup R_m$ where $Q$ is a rectangle and $R_1,\dots ,R_m$ are disjoint rectangles then ${\iint }_{Q}fdA={\sum }_{i=1}^m{\iint }_{R_i}fdA$
   Also, $\left|{\iint }_{R}fdA\right|\le {\iint }_R|f|dA$

More General Domains

Definition: A y-simple domain is the set of all points $(x,y)$ such that $x\in [a,b]$ and ${\varphi }_1(x)\le y\le {\varphi }_2(x)$ and a x-simple domain is the set of all points $(x,y)$ such that $y\in [c,d]$ and ${\psi }_1(y)\le x\le {\psi }_2(y)$. A simple region can be defined either way.

Definition: The integral over an elementary region $D\subset R$ can be defined with the regular definition of an integral over $R$ by defining a function $f^{\ast }(x,y)=\{\begin{array}{ll}f(x,y)& \text{if }(x,y)\in D\\ 0& \text{if }(x,y)\notin D\text{ and }(x,y)\in R\end{array}$. Then say ${\iint }_{D}f(x,y)dA={\iint }_R{f}^{\ast }(x,y)dA$

Theorem: If $D$ is a $y$-simple region then ${\iint }_{D}f(x,y)dA={\int }_a^b{\int }_{{\varphi }_1(x)}^{{\varphi }_2(x)}f(x,y)dydx$. If $D$ is an $x$-simple region then ${\iint }_{D}f(x,y)dA={\int }_c^d{\int }_{{\psi }_1(y)}^{{\psi }_2(y)}f(x,y)dxdy$

Then, to find the area of $D$ one could substitute $f=1$ to either of these formula. ${\int }_a^b({\varphi }_2(x)-{\varphi }_1(x))dx$

Note that sometimes the order of iterated integrals affects the difficulty of evaluating the volume.

If there are numbers $m$ and $M$ such that for all $(x,y)\in D$, $m\le f(x,y)\le M$ then $m\cdot A(D)\le {\iint }_{D}f(x,y)dA\le M\cdot A(D)$ where $A(D)$ is the area of $D$

Theorem: Suppose $f:D\to \mathbb{R}$ is continuous and $D$ is an elementary region, then for some point $(x_0,y_0)\in D$, ${\iint }_{D}f(x,y)dA=f(x_0,y_0)A(D)$

Triple Integrals

Definition: Let $f$ be a bounded function of three variables defined on $B$. The Reimann sum can be defined as $S_n={\sum }_{i=0}^{n-1}{\sum }_{j=0}^{n-1}{\sum }_{k=0}^{n-1}f({\mathbf{c}}_{ijk})\Delta V$. If ${\lim }_{n\to \infty }{S}_n=S$ exists and is independent of ${\mathbf{c}}_{ijk}$ then $f$ is integrable and $S$ is the triple integral of $f$ over $B$, denoted by ${\iiint }_{B}fdV={\iiint }_{B}f(x,y,z)dV={\iiint }_{B}f(x,y,z)dxdydz$.

The order of iterated integrals does not matter, similar to the case of double integrals.

Note that with regards to elementary regions, the notation ${\gamma }_1(x,y)\le z\le {\gamma }_2(x,y)$ is used for $z$