5.21 Riemann double integrals and areas

In the past lecture we defined iterated integrals of functions in two variables, which are a kind of double integrals, on domains that are rectangles and proved that one may compute the integral in any order if the function is continuous (Fubini's theorem).

\(f:\mathbb{R}\to \mathbb{R}\quad f(x,y)\geq0\)
\(R = [a,b] \times [c,d] \quad f\) continuous
Volume \(\displaystyle= \int_{a}^{b} \int_{c}^{d} f(x,y) dy dx = \int_{c}^{d} \int_{a}^{b} f(x,y) dx dy = \iint_{R} f(x) dA\)

In this lecture we make our first steps toward integrals over more general domains.

We consider now domains thar are not necessarily rectangles.

For example, the unit disc \(B((0,0), 1) = \{(x, y) \in \mathbb{R}^2 \mid x^2 + y^2 < 1\}\)

Its boundary is the set \(\{(x, y) \in \mathbb{R}^2 | x^2 + y^2 = 1\}\)

We have already presented a similar example when we analyzed the function \(f(x, y) = x^2 + y^2\)

If we fix a value of \(x\), then the variable \(y\) varies between the red curve and the blue curve.

In both cases, the curves are functions on \(x\):

As you see, the limits of integration are given by functions depending on \(x\).

Questions:

Is \(F(x)\) continuous on this domain? Is it differentiable?
What if the region is more complicated?

Idea: try to describe the domain by using rectangles, and then sum up all integrals over rectangles.

The intuitive idea for computing a volume below a surface is that this volume should be something like the area of the region \(S\) times the height, which is given by the function \(f(x, y)\):

We have then two possible issues:

To compute the area of \(S\);
To determine which functions \(f(x,y)\) are integrable over \(S\).

As we did in the case in one variable, we can try to approximate the volume by using partitions of regions in \(\mathbb{R}^2\)!!

As we did before, let us start with rectangles to get the idea:
Let \(R = [a,b] \times [c,d] \subset \mathbb{R}^2\) be a rectangle and \(f: R \rightarrow \mathbb{R}\) be a function.
We take a partition of both intervals \([a,b]\) and \([c,d]\) as we did in one variable.

\(\{a=x_0, x_1, ..., x_{n-1}, x_n=b\}\) and \(\{c=y_0, y_1, ..., y_{k-1}, y_k=d\}\)

In this way, we get a partition of \(R\) into small rectangles \(R_{ij} = [x_{i-1}, x_i] \times [y_{j-1}, y_j]\) for all \(1 \le i \le n, 1 \le j \le k\)
We denote this partition by \(P=\{R_{ij}\}\) and the area of each small rectangle by \(a(R_{ij}) = (x_i - x_{i-1})(y_j - y_{j-1})\)
Our intention is to approximate the volume by summing up the volumes of the columns with base the rectangles \(R_{ij}\).

Instead of introducing the definition of integrals with lower and upper sums as we did in the one-dimensional case, we do Riemann sums this time.

So, for each pair \(i, j\), we choose an evaluation point \(t_{ij} \in R_{ij}\).

Definition

The Riemann sum of \(f\) with respect to the partition \(P = \{R_{ij}\}\) and the evaluation points \(t = \{t_{ij}\}\) is the sum

\[ S(f, P, t) = \sum_{i=1}^{n}\sum_{j=1}^{k}f(t_{ij}) a(R_{ij}) \]

Here, we conclude that \(f\) is bounded and the limit of \(a(R_{ij})\) goes to 0 since if it is Riemann integrable and \(\lambda(P)<\delta\), then we have more and more interval, then \(n,k\) goes to infinity
The diameter of the partition \(P = \{R_{ij}\}\) is \(\lambda(P) = \max (\{x_i - x_{i-1} \mid 1 \le i \le n\} \cup \{y_j - y_{j-1} \mid 1 \le j \le k\})\)
Remark. If we try to define \(\lambda(P)\) as the maximum of the area of the rectangles \(R_{ij}\), this will fail, because one may have rectangles with area very small but with a side of length 1. This would produce a bad approximation of the volume we are trying to compute.

Definition

A function \(f: R \to \mathbb{R}\) is called Riemann integrable if there exists \(I \in \mathbb{R}\) such that \(\forall \varepsilon > 0\), there exists \(\delta > 0\) so that for every partition \(P = \{R_{ij}\}\) with \(\lambda(P) < \delta\) and for any choice of evaluation points \(t = \{t_{ij}\}\) for \(P\) we have \(|S(f, P, t) - I| < \varepsilon\)
In case the function \(f\) is Riemann integrable we write \(\displaystyle I = \int_R f\)

Remark

This definition is completely analogous to the definition of Riemann integrable function in the one-dimensional case.

Theorem

Let \(f\) be a Riemann integrable function in a rectangle \(R\). Then \(f\) is bounded.

Now we relate this definition with lower and upper sums, which are called Darboux sums.

Given a bounded function \(f: R \to \mathbb{R}\) and a partition \(P = \{R_{ij}\}\) of the rectangle \(R\) we define
\(M_{ij} = \sup\{f(x,y) \mid (x,y) \in R_{i,j}\}\), \(m_{ij} = \inf\{f(x,y) \mid (x,y) \in R_{i,j}\}\)

Definitions

The upper Darboux sum of the function \(f\) with respect to the partition \(P = \{R_{ij}\}\) is \(\mathcal{U}(f, P) = \sum_{i=1}^n \sum_{j=1}^k M_{ij} \alpha(R_{ij})\)
The lower Darboux sum of the function \(f\) with respect to the partition \(P = \{R_{ij}\}\) is \(\mathcal{L}(f, P) = \sum_{i=1}^n \sum_{j=1}^m m_{ij} \alpha(R_{ij})\)

As one may expect, it is possible to characterize Riemann integrable functions using upper and lower sums.

Theorem

Let \(f: R \to \mathbb{R}\) be a bounded function on a rectangle \(R\). Then \(f\) is Riemann integrable if and only if for all \(\varepsilon > 0\), there exists a partition \(P = \{R_{ij}\}\) such that \(U(f, P) - L(f, P) < \varepsilon\)

With this definition one may start computing areas of different sets in the plane.

But we have to be careful, since not all set have area!!

Definitions

Given a set \(S \subseteq \mathbb{R}^2\), the characteristic function of \(S\) is the function \(\chi_S: \mathbb{R}^2 \to \mathbb{R}\) given by
\(\chi_{S}(x, y) = \begin{cases} 1 & \text{if } (x,y) \in S \\ 0 & \text{if } (x,y) \notin S \end{cases}\)
A bounded set \(S \subseteq \mathbb{R}^2\) has area if its characteristic function \(\chi_S\) is Riemann integrable in some rectangle \(R\) containing \(S\). In case \(S\) has area, we denote it by \(\displaystyle a(S) = \int_{R} \chi_{S} = \int_{S} 1\)

Remark

The definition does not depend on the rectangle we take.

Example

\(\chi_{S}(x, y) = \begin{cases} 1 & \text{if } (x,y) \in S \\ 0 & \text{if } (x,y) \notin S \end{cases}=\begin{cases} 1 & \text{if } a\leq x\leq b,c\leq y\leq d \\ 0 & \text{if not} \end{cases}\)

\(\implies S(\chi_{R},P,t)=\sum_{i=1}^{n}\sum_{j=1}^{k}a(R_{ij})=\sum_{i=1}^{n}\sum _{j=1}^{k}(x_{i}-x_{i-1})(y_{j}-y_{j-1})=\left(\sum_{i=1}^{n}(x_{i}-x_{i-1})\right )\left(\sum_{j=1}^{k}(y_{j}-y_{j-1})\right)=(b-a)(d-c)\)

If \(S\) is a rectangle \({\displaystyle\int_{S}1=\int_{S}\chi_{S}(x,y)dA=\text{Area of rectangle}=(b-a)(d-c)} =\) \(\displaystyle\int_{a}^{b} \int_{c}^{d} \chi_{S}(x,y) dy dx\)

The last equality is because \(\displaystyle \int_{a}^{b}\int_{c}^{d}\chi_{R}(x,y) \, dy \, dx = \int_{a}^{b}\int_{c}^{d}1 \, dy \, dx=\int_{a}^{b}(d-c) \, dx=(d-c)\int_{a}^{b}1 \, dx=(d-c)(b-a)\)

The idea behind all these definitions is that this is exactly what we were looking for to compute volumes and integrals.

Let us state some properties to check that our definition coincide with our intuition.

To give another characterization of Riemann integrable functions, we introduce one of the most important notions in measure theory, the measure zero sets.

Idea: We saw that when one integrates functions in one variable, the values of the function in a finite number of points does not change the computation:\(\displaystyle \int_{a}^{b} f(x) dx = \int_{a}^{c} f(x) dx + \int_{c}^{b} f(x) dx\)

But what corresponds to extreme points in an interval in \(\mathbb{R}\) can be curves in \(\mathbb{R}^2\)!!!

Definition

A set \(T \subseteq \mathbb{R}^2\) has measure zero if \(\forall \ \varepsilon > 0\), there is a countable collection of rectangles \(\{R_i\}\) so that \(T \subset \cup_i R_i\) and \(\sum_i a(R_i) < \varepsilon\)

Example

A curve in \(D^2\) has measure zero (it means it has no area)
For \(\R^1\), a famous measure zero set is Cantor set \(\rightarrow\) Fractals

Theorem

Let \(S \subset \mathbb{R}^2\) be bounded set. Then \(S\) has area if and only if \(\partial S\) has measure zero.

Proof

\(S\) has area\(\iff\)\(\chi_S\) is Riemann integrable\(\iff|S(f, P, t) - I| < \varepsilon\)

\(S(\chi_{S},P,t)=\sum_{i=1}^{n}\sum_{j=1}^{k}\chi_{S}(t_{ij})a(R_{ij})\) where \(a(R_{ij})\) is as small as we want

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