Course:MATH200/IntegrationOnRectangles

Integration on Rectangles

Review of integration in one variable

Right Riemann sums

Let a and b be real numbers with a<b and let f:[a,b]→R be a function on the closed interval [a,b]={x∈R: a≤ x≤ b}. Let n be a positive integer, and set Δx=(b-a)/n. For each integer i with 1≤i≤n set x_i=a+iΔx. Then the n-th right-endpoint Riemann sum is the number

 $\displaystyle R_{n}(f):=\sum _{i=1}^{n}f(x_{i})\,\Delta x=f(x_{1})\Delta x+f(x_{2})\Delta x+\cdots +f(x_{n})\Delta x.$

The Riemann integral of a continuous function

Definition. Let f:[a,b]→R be a continuous function. Then we say that the Riemann integral of f on [a,b] is I∈R and write

 $\displaystyle \int _{a}^{b}f(x)\,dx=I$

if for every positive real number ε there exists a real number N such that

  $\displaystyle n\geq N\Rightarrow |R_{n}(f)-I|<\epsilon .$ 

Another way to say this is to say that

 $\displaystyle \int _{a}^{b}f(x)\,dx=\lim _{n\to \infty }R_{n}(f).$

Theorem. The Riemann integral of a continuous function always exists.

Remarks on Riemann integrals

When f is non-negative on [a,b], the usual way to think about the integral is as representing the area A under the graph of a the function. We approximate this area by partitioning the interval [a,b] into n sub-intervals of equal length I₁=[a,x₁], I₂=[x₁, x₂],..., I_n=[x_n-1,x_n]. Then we approximate f on the k-th interval I_k by its f(x_k) at the right endpoint of the sub-interval. The number f(x_k) the approximates the area under the part of the graph of the function lying over I_k, and the sum approximates the area under A.

For discontinous f the definition of the integral needs to be changed slightly because it is not enough to consider the values of f at just the right endpoints of the subintervals. It is also very convenient to consider partitions where the subintervals do not necessarily all have the same length.

Integration of two variable functions

Top Right Riemann sumbs over rectangles

Now let a,b,c,d be real numbers with a≤ b, c≤ d. The set

 $D:=[a,b]\times [c,d]=\{(x,y)\in \mathbf {R} ^{2}:a\leq x\leq b,c\leq y\leq d\}$

is simply the rectangle with vertices (a,c), (a,d), (b,c) and (b,d) and area (b-a)(d-c). Suppose n is a positive integer. Set Δ x=(b-a)/n, Δy=(d-c)/n. For i an integer between 0 and n set

 $\displaystyle x_{i}=a+i\Delta x,y_{i}=c+i\Delta y,\Delta A=\Delta x\Delta y.$

The n-th Riemann sum is the sum

 $R_{n}(f)=\sum _{i=1}^{n}\sum _{j=1}^{n}f(x_{i},y_{j})\,\Delta A$

Remark. It might be instructive to write the sum out in full. For example, suppose n=2. Then there are four terms,

 $\displaystyle R_{2}(f)=[f(x_{1},y_{1})+f(x_{1},y_{2})+f(x_{2},y_{1})+f(x_{2},y_{2})]\Delta A.$

The Riemann integral of functions on a rectangle

The definition of the Riemann integral is almost the same as before.

Definition. Let f:D→R be a continuous function. Then we say that the Riemann integral of f on [a,b] is I∈R and write

 $\displaystyle \int \!\int _{D}f(x,y)\,dA=I$

if for every positive real number ε there exists a real number N such that

  $\displaystyle n\geq N\Rightarrow |R_{n}(f)-I|<\epsilon .$ 

Another way to say this is to say that

 $\displaystyle \int \!\int _{D}f(x,y)\,dA=\lim _{n\to \infty }R_{n}(f).$

Remarks on the definition

Again this definition only works for continuous functions. For discontinuous functions it can give the wrong answer because we have only considered the value of f at the top right vertex of the rectangle D_ij=[x_i-1,x_i]x[y_j-1,y_j]. The general definition is given below.

Evaluating Integral over Rectangle

Fubini's Theorem

The following result is the key to evaluating integrals (without having to take limits).

Fubini's Theorem Let f:D→R be a continous function on a rectangle D=[a,b]x[c,d]. Define functions

 ${\begin{aligned}g(x)&:=\int _{c}^{d}f(x,y)\,dy\\h(y)&:=\int _{a}^{b}f(x,y)\,dx.\end{aligned}}$

Then g(x) is a continuous function on [a,b] and h(y) is a continuous function on [c,d]. Moreover,

 $\displaystyle \int _{D}f(x,y)\,dA=\int _{a}^{b}g(x)\,dx=\int _{c}^{d}h(y)\,dy.$

Remarks on Fubini

Another more compact way to state the theorem is to say that

 $\displaystyle \int \!\int _{D}f(x,y)\,dA=\int _{a}^{b}\int _{c}^{d}f(x,y)\,dy\,dx=\int _{c}^{d}\int _{a}^{b}f(x,y)\,dx\,dy.$

We will not cover the proof of the theorem. However, we do have

  $\displaystyle R_{n}(f)=\sum _{i=1}^{n}\sum _{j=1}^{n}f(x_{i},y_{j})\,\Delta y_{j}\Delta x_{i}=\sum _{j=1}^{n}\sum _{i=1}^{n}f(x_{i},y_{j})\,\Delta x_{i}\Delta y_{j}.$

(If you are at all uncertain about this, write out the sum for n=2 or 3.) So on the level of Riemann sums (before we take the limit) the theorem holds.

For function which are non-negative, the integral

 $\displaystyle \int \!\int _{D}f(x,y)\,dA$

is supposed to represent the volume of the region in space lying under the graph of the function and over the rectangle D. Fubini's theorem then says that this volume can be calculated by the usual method from one variable calculus: first calculate the areas g(x₀) the cross-sections of the region with x=x₀, then integrate these cross-sectional areas g(x) as x goes from a to b.

Examples of Integration on rectangles

Example 1.

Problem. Let D=[0,1]x[0,1]. Compute

 $\displaystyle \int \!\int _{D}{\frac {x}{1+xy}}\,dA.$

Solution. By Fubini, we have

${\begin{aligned}\int \!\int _{D}{\frac {x}{1+xy}}\,dA&=\int _{0}^{1}\!\int _{0}^{1}{\frac {x}{1+xy}}\,dx\,dy\\&=\int _{0}^{1}[\ln(1+xy)]_{x=0}^{x=1}\,dy\\&=\int _{0}^{1}\ln(1+y)\,dy\\&=\int _{1}^{2}\ln u\,du,\quad u=1+y\\&=[u\ln u-u]_{1}^{2}\\&=2\ln 2-1.\end{aligned}}$

where we did the last integral by integrating by parts.

More General Riemann Sums

To be able to define integration of discontinuous functions, we need to consider more general Riemann Sums.

Partitions of Rectangles with sample points

Let D=[a,b]x[c,d] be a rectangle and let n and m be positive integers. Define Δx=(b-a)/n and Δy=(d-c)/m. Then set

 ${\begin{aligned}x_{i}&=a+i\Delta x;i=0,....,n\\y+j&=c+j\Delta y;j=0,...,m.\\\end{aligned}}$

Set D_ij=[x_i-1,x_i]x,y_j-1,y_j]. D_ij is called the (i,j)-th rectangle in the partition of D into nm subrectangles. The area of D_ij is &Delta A;:=ΔxΔy.

The choice of a point $\displaystyle t_{i}j=(x_{ij}^{*},y_{ij}^{*})\in D_{ij}$ for each 1≤ i≤ n, 1≤j≤m is called the choice of a sample point . It is also called a tagging of the partition of D. The data of D with n,m and the choice of the t_ij is sometimes called a tagged partition.

Riemann sums associated to partition with sample points

Lef f:D→R be a function.

Definition. The number

 $\displaystyle R_{nm}(f)=\sum _{i=1}^{n}\sum _{j=1}^{m}f(t_{ij})\Delta A$

is called the Riemann sum associated to the tagged partition above.

=Common choices of sample points

1. top-right: t_ij=(x_i,y_j).
2. midpoint: $\displaystyle t_{ij}=({\bar {x}}_{ij},{\bar {y}}_{ij});{\bar {x}}_{i}=(x_{i-1}+x_{i})/2,{\bar {y}}_{j}=(y_{j-1}+y_{j})/2.$

The Riemann Integral for General functions

We say that $\int _{D}f\,dA=I$ if for every positive real number ε there exists a real number N such that

 $\displaystyle |R_{nm}f-I|<\epsilon$

for any tagged partition with n,m>N.