Q 12. - Calculus

Question

Which of the iterated integrals in Exercises 9–12 could correctly be used to evaluate the double integral ${\iint }_{0}f(x,y)dA$

${\int }_2^0{\int }_{-y+2}^{0}f(x,y)dxdy$

Step-by-Step Solution

Answer

${\int }_0^2{\int }_0^{-y+2}f(x,y)dxdy$ will give the correct value of the given integral.

Step 1: Given Information

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

Step 2: Consideration

Consider iterated integral

${\int }_2^0{\int }_{-y+2}^{0}f(x,y)dxdy$

Region is triangular and can be of any type.

For type II region, inner integral is solved wrt $x$.

This is not true for iterated integral ${\int }_2^0{\int }_{-y+2}^{0}f(x,y)dxdy$

Step 3: Simplification

Keeping the order of integral same, limits of both inner and outer integral should be reversed.

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

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