Question

Write one or more integrals using the order of integration dxdy equivalent to f(x, y) dydx​

Answer 1

The lines $y=\frac x2$ and $y=3-x$ meet when

$\dfrac x2 = 3-x \implies \dfrac{3x}2 = 3 \implies x=2$

at which point $y=\frac22 = 1$. They also have $y$-intercepts of $y=0$ and $y=3$, respectively.

Split the region up at $y=1$. Then the integral of $f$ in the opposite order of variables is

$\displaystyle \int_0^1 \int_0^{2y} f(x,y) \, dx \, dy + \int_1^3 \int_0^{3-y} f(x,y) \, dx \, dy$