Question

Which of the iterated integrals in Exercises below could correctly

be used to evaluate the double integral:

${\int }_4^1{\int }_6^{2}f(x,y)dydx$

Short answer

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

Step-by-step solution

Given Information

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

Simplification

In iterated integral ${\int }_4^1{\int }_6^{2}f(x,y)dydx$, inner integral is w.r.t $y$and limits are of $y$of inner integral.

This is true for $x$also, the limits are reversed.

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

The region is bounded by $y=2$below, $y=6$above, $x=1$to left and $x=4$to right.

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