Answer
The estimate of the double integral of $f$ over the small rectangle ${\cal R} = \left[ {0.9,1.1} \right] \times \left[ {1.9,2.1} \right]$ is $0.16$.
Work Step by Step
We are given $f\left( {1,2} \right) = 4$.
Let the partition be regular and the dimensions of the subrectangle be $\Delta x = 0.2$ and $\Delta y = 0.2$ (please see the figure attached). Since the region is ${\cal R} = \left[ {0.9,1.1} \right] \times \left[ {1.9,2.1} \right]$, so the grid $N \times M$ is $1 \times 1$.
The Riemann sum ${S_{1,1}}$ is the estimate of the double integral $\mathop \smallint \limits_{x = 0.9}^{1.1} \mathop \smallint \limits_{y = 1.9}^{2.1} f{\rm{d}}y{\rm{d}}x$, that is
${S_{1,1}} = \mathop \sum \limits_{i = 1}^1 \mathop \sum \limits_{j = 1}^1 f\left( {{P_{ij}}} \right)\Delta {x_i}\Delta {y_j} = 4\cdot0.2\cdot0.2 = 0.16$
So, the estimate of the double integral of $f$ over the small rectangle ${\cal R} = \left[ {0.9,1.1} \right] \times \left[ {1.9,2.1} \right]$ is $0.16$.
