Q 6. - Calculus

Question

Which of the iterated integrals in Exercises below could correctly

be used to evaluate the double integral:

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

Step-by-Step Solution

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 1: Given Information

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

Step 2: 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 ${\int }_1^4{\int }_2^{6}f(x,y)dydx$