Q 7. - Calculus

Question

Which of the iterated integrals in Exercises below could correctly

be used to evaluate the double integral:

${\int }_1^4{\int }_2^{6}f(x,y)dxdy$

Step-by-Step Solution

Answer

The integral that will give correct value of ${\iint }_{R}f(x,y)dA$ is ${\int }_1^4{\int }_2^{6}f(x,y)dydx$

Step 1: Given Information

Rectangular region is bounded by $x=1,x=4,y=2\text{ and }y=6$

Step 2: Consideration

Consider integral

${\int }_1^4{\int }_2^{6}f(x,y)dxdy$

The inner integral is wrt $x$ and limits of inner integral are of $y$ .

Same is applicable for $x$.

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

Step 3: Simplification

If region is considered of type I keeping limits same, order of integration should be same.

Inner integral is wrt $y$.

Hence integral that will give correct value of ${\iint }_{R}f(x,y)dA$ is ${\int }_1^4{\int }_2^{6}f(x,y)dydx$