## Related questions with answers

In given problem, $R=\{(x, y): 0 \leq x \leq 6,0 \leq y \leq 4\}$ and P is the partition of R into six equal squares by the lines x=2, x=4, and y=2. Approximate $\iint_R f(x, y) d A$ by calculating the corresponding Riemann sum $\sum_{k=1}^6 f\left(\bar{x}_k, \bar{y}_k\right) \Delta A_k$, assuming that $\left(\bar{x}_k, \bar{y}_k\right)$ are the centers of the six squares.

$f(x, y)=\frac{1}{6}(48-4 x-3 y)$

Solution

VerifiedConsider the rectangle $R=\left\{\left(x,y\right):0\leq x\leq 6,0\leq y\leq 4\right\}$ and let $f:R\rightarrow\mathbb R$ de defined by

$f\left(x,y\right)=\displaystyle\frac{1}{6}\left(48-4x-3y\right).$

Below is a sketch of the rectangle $R$ partitioned in six squares by the lines $x=2,x=4$ and $y=2$. For each $1\leq k\leq 6$ we have labeled one of the squares as $R_k$.

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