Question

What does the Midpoint Rule for double integrals say?

Short answer

The required answer is \(\iint_{R}{{}}f(x,y)dA=\sum\limits_{i=1}^{m}{{}}\sum\limits_{j=1}^{n}{{}}fxi,yj\)

Step-by-step solution

Definition

The Midpoint Rule for double integrals gives us the following approximation:

\(\iint_{R}{{}}f(x,y)dA=\sum\limits_{i=1}^{m}{{}}\sum\limits_{j=1}^{n}{{}}fxi,yj\)

Represent area of subrectangle

It's important to note that \({{x}^{^{\_}}}\) is the middle point of the interval \(\left( {{x}_{i-1}},{{x}_{i}} \right)\) and \({{y}^{^{\_}}}\) the middle point of the interval \(\left( {{y}_{j-1}},{{y}_{j}} \right)\) and \(\Delta A\) represents the area of one subrectangle.

Hence, the required answer is \(\iint_{R}{{}}f(x,y)dA=\sum\limits_{i=1}^{m}{{}}\sum\limits_{j=1}^{n}{{}}fxi,yj\)