ahmet 9 Posted July 16 (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 July 16 by ahmet latex formula,correction on mathematical terms 0 Share this post Link to post Share on other sites

studiot 1980 Posted July 16 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 ( α≤x≤β and a≤y≤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 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 0 Share this post Link to post Share on other sites

ahmet 9 Posted July 16 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. 0 Share this post Link to post Share on other sites

HallsofIvy 5 Posted July 19 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). 0 Share this post Link to post Share on other sites