Question

Which of the iterated integrals in Exercises 9–12 could correctly be used to evaluate the double integral${\iint }_{\Omega }f(x,y)dA,$

${\int }_0^2{\int }_0^{-y+2}f(x,y)dxdy$

Short answer

The iterated integral${\int }_0^2{\int }_0^{-y+2}f(x,y)dxdy$ will give correct value of${\iint }_{\Omega }f(x,y)dA$

Step-by-step solution

Given Information

Rectangular region is bounded below by $x$axis, left by $y$axis and along the line$y=-x+2$

Simplification

Consider iterated integral

${\int }_0^2{\int }_0^{-y+2}f(x,y)dxdy$

Region is triangular and can be of either type

Inner integral is solved wrt $x$for type II.

In ${\int }_0^2{\int }_0^{-y+2}f(x,y)dxdy$also, inner integral is solved wrt $x$

Hence, it will give correct value of${\iint }_{\Omega }f(x,y)dA$