I am having issues switching an iterated integral from dxdy to dydx:
Switch the integral $\int_0^2 \int_y^{2y}6xy$ dx dy to a dy dx integral in the form of $\int_0^2 \int_{...}^{...}6xydydx$ + $\int_2^4 \int_{...}^{...}6xydydx$
What I'm having issues with here is trying to break it up into pieces-- is the first one just from 0 to y, and the next one from y to 2y?
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$\begingroup$The domain of integration is the inside of the triangle with vertices at the points $(0,0)$, $(2,2)$ and $(4,2)$, bounded by the lines:$y=x$, $y=x/2$ and $y=2$. The integral of an arbitrary function $f(x,y)$ in this domain can be done first in $x$ and then in $y$, or first in $y$ and then in $x$:
$$\int_0^2dy\int_y^{2y}f(x,y)\,dx = \int_0^2dx\int_{x/2}^xf(x,y)\,dy+\int_2^4dx\int_{x/2}^2f(x,y)\,dy$$
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