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Given $\iint_R|xy-1|\,dA$, where $R= [0,1] \times [0,1]$ to deal with this questions, I always sub $R$ into the formula |xy-1|: $\int _0^1 \int _0^1\:|xy-1|\,dx\,dy.$

But what is the actual meaning of $R= [0,1] \times [0,1]$? Is it a dot product?

md2perpe
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oscar
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    Is the Cartesian product – daniel Jun 01 '24 at 13:16
  • As R= [0,1] x [0,1],is R = 1 after dot product? – oscar Jun 01 '24 at 13:17
  • The meaning of $I=[0,1]$ is the closed interval, as a subset of the real numbers. Then $R=I\times I$ is the Cartesian product, as a subset of $\Bbb R \times \Bbb R$. Where do you see a dot product, or even after dot product? – Dietrich Burde Jun 01 '24 at 13:17
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    Given two sets $A$ and $B$, $A\times B$ is the set of the pairs $(x,y)$ such that $x\in A$ and $y\in B$. You'll find it as "Cartesian product" in the literature. The set-theoretic construction has some details but for practical purposes you may identitfy an $n$-tuple $(x_1,\cdots, x_n)\in S_1\times\cdots\times S_n$ with a function $x:{1,\cdots,n}\to \bigcup_{i=1}^n S_i$ such that, for all $i$, $x(i)=x_i\in S_i$. So in your case $[0,1]\times [0,1]$ would be the set of the pairs $(x_1,x_2)$ of real numbers such that $0\le x_1\le 1\land 0\le x_2\le 1$. – Sassatelli Giulio Jun 01 '24 at 13:20
  • o thankyou,i am a newbie in calculus :( – oscar Jun 01 '24 at 13:24
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    @oscar more of an elementary set theory thing, will be covered in any "intro to proof-based math" or similar – George Coote Jun 01 '24 at 13:27
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    You don't even need an "intro to proof-based math" to find discussions of intervals and Cartesian products, that should be covered in most precalculus texts. – Lee Mosher Jun 01 '24 at 13:30
  • What textbook are you using? I'm asking because I'm not sure if I've ever seen an elementary calculus text use Cartesian product notation (and most anything else, including union, intersection, etc.) without indicating somewhere the notation -- in a "foreword or preface", in a list of symbols either at the front of the book or the back of the book, in an early section or an appendix dealing with background material and notation, at the beginning of a chapter or section where the notation is first used, etc. I also agree with @Lee Mosher, but stand by my previous sentence. – Dave L. Renfro Jun 01 '24 at 13:35
  • Your textbook should define this notation. Check the introductory section on multiple integration. It should be in there. – Cameron L. Williams Jun 01 '24 at 16:49
  • @LeeMosher I have not seen any precalculus text mention Cartesian products. Intervals, yes. But set operations other than union? No. – Sean Roberson Jun 01 '24 at 16:53
  • Here's a recent question about ordered pairs and Cartesian products: https://math.stackexchange.com/q/4924488/168433 – md2perpe Jun 01 '24 at 16:59
  • @Sean Roberson: I looked at 5 or 6 precalculus/college-algebra texts on my shelves (published after 1990, so definitely not biased towards 1960s new math topics), and I didn't see any mention of Cartesian products, so perhaps my earlier comment "I agree with ..." was made too much in haste. However, Cartesian products are mentioned in every calculus text I looked at (but I restricted myself to those published after 1959). (continued) – Dave L. Renfro Jun 01 '24 at 21:12
  • Of course, if we go back further to the 1960s era precalculus textbooks, then Cartesian products are fairly standard -- they are even defined in my high school Algebra 2 text (beginning of Chapter 14), which was used for the course taken before taking precalculus. Note how much attention is given to set theoretic constructions in Prelude to Calculus and Linear Algebra by John M. H. Olmsted – Dave L. Renfro Jun 01 '24 at 21:12

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I'm going to expand Sassatelli Giulio's comment into an answer, perhaps one that is accessible.

Given two sets $A$ and $B$, their Cartesian product $A \times B$ is the set of ordered pairs $(a, b)$ where the first entry comes from $A$ and the second from $B$. In set-builder notation,

$$ A \times B = \{ (a, b) \ : a \in A, \ b \in B \}.$$

So the plane we're used to, $\mathbb{R}^2$, can be written as $\mathbb{R} \times \mathbb{R}.$ In your case, the region you're integrating over $R$ (not the blackboard bold one from earlier!) is the set of points $(x, y)$, where each coordinate is chosen from the interval $[0, 1].$

Take note, however, that the symbol $\times$ may mean the cross product or other things based on context, and it's important to keep that in mind.

As Dave Renfro says, it may help to always refer to the index, list of symbols, appendix, or other similar sections in your text. I have to do this many times, even in advanced texts.

Sean Roberson
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