# 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:

$\int^{v(x)}_{y=u(x)} f(x,y)dy$ if in this integral $f(x,y)$ function  ( $\alpha \leq x \leq \beta$ and $a \leq y \leq b$ ) 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

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

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52 minutes ago, studiot said:

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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