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Hi, everyone!

I have a problem in understading the change of variables in double integrals. Here is an example

[tex]\int\int x^2+y^2dx dy=\int \frac{x^3}{3}+y^2x dy=\frac{x^3y}{3}+\frac{y^3x}{3}+C_1[/tex]

but if I first do a change in poral coordinates I get

[tex]\int\int r^2 r drd\theta=\int\frac{r^4}{4}d\theta=\frac{r^4\theta}{4}=\frac{1}{4}(x^2+y^2)^2 arctan(\frac{y}{x})+C_2[/tex]

which is not the first answer. A more simple example is

[tex]\int\int dxdy=xy+C_1[/tex]

and in poral

[tex]\int\int rdrd\theta=\int\frac{r^2}{2}d\theta=\frac{r^2\theta}{2}=\frac{1}{2}(x^2+y^2) arctan(\frac{y}{x})+C_2[/tex]

Can someone explain what I do wrong?

Thanks in advance!

I have a problem in understading the change of variables in double integrals. Here is an example

[tex]\int\int x^2+y^2dx dy=\int \frac{x^3}{3}+y^2x dy=\frac{x^3y}{3}+\frac{y^3x}{3}+C_1[/tex]

but if I first do a change in poral coordinates I get

[tex]\int\int r^2 r drd\theta=\int\frac{r^4}{4}d\theta=\frac{r^4\theta}{4}=\frac{1}{4}(x^2+y^2)^2 arctan(\frac{y}{x})+C_2[/tex]

which is not the first answer. A more simple example is

[tex]\int\int dxdy=xy+C_1[/tex]

and in poral

[tex]\int\int rdrd\theta=\int\frac{r^2}{2}d\theta=\frac{r^2\theta}{2}=\frac{1}{2}(x^2+y^2) arctan(\frac{y}{x})+C_2[/tex]

Can someone explain what I do wrong?

Thanks in advance!

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