Q 5. - Calculus

Question

Which of the iterated integrals in Exercises below could correctly

be used to evaluate the double integral:

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

Step-by-Step Solution

Answer

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

Step 1: Given Information

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

Step 2: Definition of integral region

Consider an iterated integral

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

The region is rectangular with is sides parallel to $xy$ plane.

It can be of either type.

For type I , the integration is first done with respect to $y$ and then $x$.

The inner integral to be solved wrt $y$.

Step 3: Simplification

In ${\int }_1^4{\int }_2^{6}f(x,y)dydx$, inner integral is solved wrt $y$ and integral is solved wrt $x$

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