Q 9. - Calculus

Question

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

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

Step-by-Step Solution

Answer

The iterated integral will give correct value of ${\iint }_{\Omega }f(x,y)dA$

Step 1: Given Information

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

Step 2: Simplification

Consider an iterated integral ${\int }_0^2{\int }_0^{-x+2}f(x,y)dydx$

The region is triangular region, can be of any type.

For type I region, inner integral is wrt $y$

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

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