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

`\frac x2 = 3-x \implies \frac{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`