I'm trying to understand why I get the same answer when I perform a double integral of a simple

function of x and y over specific ranges.

The function is f(x,y)=3*x*y.

In case one, I first integrate f(x,y) from 0 to x with respect to y. I then integrate that result from 0 to 3 with respect to x.

. . . . .[tex]\displaystyle \int_0^3\, \int_0^x\, 3xy\, dy\, dx[/tex]

In the second case, I first integrate f(x,y) from y to 3 with respect to x. I then integrate that result from 0 to 3 with respect to y.

. . . . .[tex]\displaystyle \int_0^3\, \int_y^3\, 3xy\, dx\, dy[/tex]

The answer is the same in both cases (30.375). Can anyone help me with an insight into a conceptual understanding of why that is?

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