First we examine which values of the area of rectangles are acceptable:
Since, $0 < x, y \le 8$, the integer $40$ is written only as $40 = 5 \cdot 8$. Since a $5 \times 8$ rectangle can be put in the $8 \times 8$ rectangle with $4$ ways horizontally and with $4$ ways vertically we have totally $8$ such rectangles.
The numbers $39, 38, 37, 34, 33, 31$ are not the product of two integers $x, y$ with $0 < x, y \le 8$.
The number $36$ can be written uniquely $36 = 6 \cdot 6$. Since a $6 \times 6$ rectangle can be put in the $8 \times 8$ rectangle with $3^2$ ways, we have $9$ such rectangles.
The number $35$ can be written uniquely $35 = 5 \cdot 7$. Since a $5 \times 7$ rectangle can be put in the $8 \times 8$ rectangle with $2 \cdot (4 \cdot 2) = 16$ ways, we have $9$ such rectangles.
The number $32$ can be written uniquely $32 = 4 \cdot 8$. A $4 \times 8$ rectangle can be put in the $8 \times 8$ rectangle with $2 \cdot 5 = 10$ ways.
Finally, we have $8 + 9 + 16 + 10 = 43$ rectangles with the required properties.