Q 11. - Calculus

Question

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

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

Step-by-Step Solution

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 1: Given Information

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

Step 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 ${\iint }_{\Omega }f(x,y)dA$