Let $\mathcal{R}$ denote the circular region in the $xy-$plane bounded by the circle $x^2+y^2=36$. The lines $x=4$ and $y=3$ divide $\mathcal{R}$ into four regions $\mathcal{R}_i ~ , ~i=1,2,3,4$. If $\mid \mathcal{R}_i \mid$ denotes the area of the region $\mathcal{R}_i$ and if $\mid \mathcal{R}_1 \mid >$ $\mid \mathcal{R}_2 \mid >$ $\mid \mathcal{R}_3 \mid > $ $\mid \mathcal{R}_4 \mid $, determine $\mid \mathcal{R}_1 \mid $ $-$ $\mid \mathcal{R}_2 \mid $ $-$ $\mid \mathcal{R}_3 \mid $ $+$ $\mid \mathcal{R}_4 \mid $.
1. **Identify the circle and lines:**
   The circle is given by the equation \(x^2 + y^2 = 36\), which has a radius of 6. The lines \(x = 4\) and \(y = 3\) intersect the circle and divide it into four regions.

2. **Find the points of intersection:**
   - The line \(x = 4\) intersects the circle at points where \(x = 4\). Substituting \(x = 4\) into the circle's equation:
     \[
     4^2 + y^2 = 36 \implies 16 + y^2 = 36 \implies y^2 = 20 \implies y = \pm \sqrt{20} = \pm 2\sqrt{5}
     \]
     So, the points of intersection are \((4, 2\sqrt{5})\) and \((4, -2\sqrt{5})\).

   - The line \(y = 3\) intersects the circle at points where \(y = 3\). Substituting \(y = 3\) into the circle's equation:
     \[
     x^2 + 3^2 = 36 \implies x^2 + 9 = 36 \implies x^2 = 27 \implies x = \pm \sqrt{27} = \pm 3\sqrt{3}
     \]
     So, the points of intersection are \((3\sqrt{3}, 3)\) and \((-3\sqrt{3}, 3)\).

3. **Divide the circle into regions:**
   The lines \(x = 4\) and \(y = 3\) divide the circle into four regions. We need to determine the areas of these regions and their relationships.

4. **Calculate the areas of the regions:**
   - The total area of the circle is:
     \[
     \pi \times 6^2 = 36\pi
     \]

   - The lines \(x = 4\) and \(y = 3\) divide the circle into four regions. We can use symmetry and the fact that the lines divide the circle into four regions of different areas.

5. **Determine the areas of the regions:**
   - The largest region \(\mathcal{R}_1\) is the one that includes the origin and extends to the boundary of the circle.
   - The second largest region \(\mathcal{R}_2\) is adjacent to \(\mathcal{R}_1\) but does not include the origin.
   - The third largest region \(\mathcal{R}_3\) is adjacent to \(\mathcal{R}_2\) and is smaller.
   - The smallest region \(\mathcal{R}_4\) is the one that is opposite to \(\mathcal{R}_1\).

6. **Calculate the required expression:**
   We need to find \(\mid \mathcal{R}_1 \mid - \mid \mathcal{R}_2 \mid - \mid \mathcal{R}_3 \mid + \mid \mathcal{R}_4 \mid\).

   By symmetry and the properties of the circle, we can see that:
   \[
   \mid \mathcal{R}_1 \mid + \mid \mathcal{R}_4 \mid = \mid \mathcal{R}_2 \mid + \mid \mathcal{R}_3 \mid
   \]
   Therefore:
   \[
   \mid \mathcal{R}_1 \mid - \mid \mathcal{R}_2 \mid - \mid \mathcal{R}_3 \mid + \mid \mathcal{R}_4 \mid = 0
   \]

The final answer is \(\boxed{0}\)