A cubic funtion $ y \equal{} ax^3 \plus{} bx^2 \plus{} cx \plus{} d\ (a\neq 0)$ intersects with the line $ y \equal{} px \plus{} q$ at $ x \equal{} \alpha ,\ \beta ,\ \gamma\ (\alpha < \beta < \gamma).$ Find the area of the region bounded by these graphs in terms of $ a,\ \alpha ,\ \beta ,\ \gamma$.
1. The area between the cubic function \( y = ax^3 + bx^2 + cx + d \) and the line \( y = px + q \) from \( x = \alpha \) to \( x = \gamma \) is given by the integral of the difference of the two functions:
   \[
   \text{Area} = \left| \int_{\alpha}^{\gamma} \left( ax^3 + bx^2 + cx + d - (px + q) \right) \, dx \right|
   \]

2. Simplify the integrand:
   \[
   ax^3 + bx^2 + cx + d - (px + q) = ax^3 + bx^2 + (c - p)x + (d - q)
   \]

3. Given that the cubic function intersects the line at \( x = \alpha, \beta, \gamma \), we can factor the cubic polynomial as:
   \[
   ax^3 + bx^2 + cx + d - (px + q) = a(x - \alpha)(x - \beta)(x - \gamma)
   \]

4. Substitute this factorization into the integral:
   \[
   \text{Area} = \left| \int_{\alpha}^{\gamma} a(x - \alpha)(x - \beta)(x - \gamma) \, dx \right|
   \]

5. To evaluate the integral, we use the fact that the polynomial \( (x - \alpha)(x - \beta)(x - \gamma) \) is symmetric and can be integrated using standard techniques. The integral of a cubic polynomial over its roots can be simplified using properties of definite integrals and symmetry:
   \[
   \int_{\alpha}^{\gamma} (x - \alpha)(x - \beta)(x - \gamma) \, dx = \frac{1}{12} (\gamma - \alpha)^3 (2\beta - \gamma - \alpha)
   \]

6. Therefore, the integral becomes:
   \[
   \int_{\alpha}^{\gamma} a(x - \alpha)(x - \beta)(x - \gamma) \, dx = a \cdot \frac{1}{12} (\gamma - \alpha)^3 (2\beta - \gamma - \alpha)
   \]

7. Taking the absolute value and multiplying by \( a \):
   \[
   \text{Area} = \left| a \cdot \frac{1}{12} (\gamma - \alpha)^3 (2\beta - \gamma - \alpha) \right| = \frac{|a|}{12} (\gamma - \alpha)^3 |2\beta - \gamma - \alpha|
   \]

The final answer is \(\boxed{\frac{|a|}{12} (\gamma - \alpha)^3 |2\beta - \gamma - \alpha|}\)