Q 10. - Calculus

Question

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

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

Step-by-Step Solution

Answer

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

Step 1: Given Information

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

Step 2: 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.

Step 3: 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 ${\int }_0^2{\int }_0^{-x+2}f(x,y)dydx$