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ahmet

regular region

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Posted (edited)

is there such a definition in the content of integral account/calculation courses or in the content of calculus?

I remember something like this:

[math]  \int^{v(x)}_{y=u(x)} f(x,y)dy  [/math] if in this integral [math]f(x,y)[/math] function  ( [math]  \alpha  \leq x \leq \beta    [/math] and [math] a \leq y \leq b  [/math] ) is derivable in the D region that characterized with the given inequalites in the paranthesis,then this region would be called as "regular region"

but I am not sure about the exact definition

could someone provide some more context about regular region (if possible)?

thanks

 

 

 

Edited by ahmet
latex formula,correction on mathematical terms

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6 hours ago, ahmet said:

is there such a definition in the content of integral account/calculation courses or in the content of calculus?

I remember something like this:

v(x)y=u(x)f(x,y)dy  if in this integral f(x,y) function  ( αxβ and ayb ) is derivable in the D region that characterized with the given inequalites in the paranthesis,then this region would be called as "regular region"

but I am not sure about the exact definition

could someone provide some more context about regular region (if possible)?

thanks

 

 

 

Not quite sure about your statement of the integral.

What are you doing about x (a varaible) in the integral of f(x, y) dy ?

 

Is this perhaps an attempt at the REgularity Theorem in the Calculus of Variations ?

 

See page 112 here

https://www.math.uni-leipzig.de/~miersemann/variabook.pdf

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52 minutes ago, studiot said:

Not quite sure about your statement of the integral.

What are you doing about x (a varaible) in the integral of f(x, y) dy ?

 

 

 

x is a parameter here.

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If I remember correctly, a region in $R^2$ is "regular" if its boundary is a simple closed curve.  (And a curve is "simple" if it does not cross itself.)  Yes, the rectangular region $a\le x\le b$, $c\le y \le d$ is a "regular region".  As to the integral, $\int_{u(x)}^{v(x)} f(x,y)dy$, Assuming that f is an "integrable" function of y, then $\int_{u(x)}^{v(x)} f(x,y)dy$ is a function of x, F(x).    I might think of the "x" as a "parameter" in the integral but as a variable in f(x,y) and in F(x).

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