In quadrilateral $ABCD$, diagonals $AC$ and $BD$ intersect at $O$. If the area of triangle $DOC$ is $4$ and the area of triangle $AOB$ is $36$, compute the minimum possible value of the area of $ABCD$.
1. Given that diagonals \(AC\) and \(BD\) of quadrilateral \(ABCD\) intersect at \(O\), we know that the area of triangle \(DOC\) is 4 and the area of triangle \(AOB\) is 36.
2. We need to find the minimum possible value of the area of quadrilateral \(ABCD\).

To solve this, we will use the fact that the area of a quadrilateral can be divided into four triangles by its diagonals. Specifically, the area of quadrilateral \(ABCD\) can be expressed as the sum of the areas of triangles \(AOB\), \(BOC\), \(COD\), and \(DOA\).

3. Let the areas of triangles \(BOC\) and \(DOA\) be denoted as \(x\) and \(y\) respectively. Therefore, the total area of quadrilateral \(ABCD\) is:
   \[
   \text{Area of } ABCD = \text{Area of } AOB + \text{Area of } BOC + \text{Area of } COD + \text{Area of } DOA
   \]
   \[
   \text{Area of } ABCD = 36 + x + 4 + y
   \]

4. To find the minimum possible value of the area of \(ABCD\), we need to minimize \(x + y\). Since the areas of triangles \(BOC\) and \(DOA\) must be non-negative, the minimum value of \(x + y\) is 0. However, this would imply that points \(B\) and \(D\) coincide, which is not possible in a quadrilateral.

5. Therefore, the minimum non-zero values for \(x\) and \(y\) must be such that the sum of the areas of the triangles \(BOC\) and \(DOA\) is at least the sum of the areas of triangles \(DOC\) and \(AOB\). This is because the areas of the triangles formed by the diagonals must balance out to form a valid quadrilateral.

6. Given that the areas of triangles \(DOC\) and \(AOB\) are 4 and 36 respectively, the minimum possible value for the sum of the areas of triangles \(BOC\) and \(DOA\) is:
   \[
   x + y \geq 4 + 36 = 40
   \]

7. Therefore, the minimum possible value of the area of quadrilateral \(ABCD\) is:
   \[
   \text{Area of } ABCD = 36 + 4 + 40 = 80
   \]

Conclusion:
\[
\boxed{80}
\]