Question

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

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

Short answer

The iterated integral that will give correct value of integral is${\int }_0^2{\int }_0^{-x+2}f(x,y)dydx$

Step-by-step solution

Given Information

It is given that region is bounded on left by$y$axis and on left by$x$axis.

Consideration

Consider iterated integral

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

Region is triangular and can be of any type.

The inner integral is solved wrt $y$for type I.

Simplification

But in iterated integral ${\int }^{-x+2}{\int }^{2}f(x,y)dxdy$, inner integral is wrt $y$

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

The limits will be interchanged and hence the correct integral is${\int }_0^2{\int }_0^{-x+2}f(x,y)dydx$