A cubic function $ y \equal{} ax^3 \plus{} bx^2 \plus{} cx \plus{} d\ (a\neq 0)$ touches a line $ y \equal{} px \plus{} q$ at $ x \equal{} \alpha$ and intersects $ x \equal{} \beta \ (\alpha \neq \beta)$.
Find the area of the region bounded by these graphs in terms of $ a,\ \alpha ,\  \beta$.
1. Given the cubic function \( y = ax^3 + bx^2 + cx + d \) and the line \( y = px + q \), we know that the cubic function touches the line at \( x = \alpha \) and intersects it at \( x = \beta \) where \( \alpha \neq \beta \).

2. Since the cubic function touches the line at \( x = \alpha \), the function values and their first derivatives must be equal at \( x = \alpha \). Therefore, we have:
   \[
   ax^3 + bx^2 + cx + d = px + q \quad \text{at} \quad x = \alpha
   \]
   \[
   3ax^2 + 2bx + c = p \quad \text{at} \quad x = \alpha
   \]

3. The area \( A \) between the cubic function and the line from \( x = \alpha \) to \( x = \beta \) is given by:
   \[
   A = \left| \int_{\alpha}^{\beta} \left( ax^3 + bx^2 + cx + d - (px + q) \right) \, dx \right|
   \]

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

5. Since the cubic function touches the line at \( x = \alpha \), we can write the cubic function in the form:
   \[
   y = a(x - \alpha)^2(x - \beta)
   \]
   This form ensures that the cubic function has a double root at \( x = \alpha \) and a single root at \( x = \beta \).

6. The area \( A \) can now be written as:
   \[
   A = \left| \int_{\alpha}^{\beta} a(x - \alpha)^2(x - \beta) \, dx \right|
   \]

7. Expanding the integrand:
   \[
   a(x - \alpha)^2(x - \beta) = a(x^3 - (2\alpha + \beta)x^2 + (\alpha^2 + 2\alpha\beta)x - \alpha^2\beta)
   \]

8. Integrating term by term:
   \[
   \int_{\alpha}^{\beta} a(x^3 - (2\alpha + \beta)x^2 + (\alpha^2 + 2\alpha\beta)x - \alpha^2\beta) \, dx
   \]
   \[
   = a \left[ \frac{x^4}{4} - (2\alpha + \beta) \frac{x^3}{3} + (\alpha^2 + 2\alpha\beta) \frac{x^2}{2} - \alpha^2\beta x \right]_{\alpha}^{\beta}
   \]

9. Evaluating the definite integral:
   \[
   = a \left( \left[ \frac{\beta^4}{4} - (2\alpha + \beta) \frac{\beta^3}{3} + (\alpha^2 + 2\alpha\beta) \frac{\beta^2}{2} - \alpha^2\beta \beta \right] - \left[ \frac{\alpha^4}{4} - (2\alpha + \beta) \frac{\alpha^3}{3} + (\alpha^2 + 2\alpha\beta) \frac{\alpha^2}{2} - \alpha^2\beta \alpha \right] \right)
   \]

10. Simplifying the expression:
    \[
    = a \left( \frac{\beta^4}{4} - \frac{(2\alpha + \beta) \beta^3}{3} + \frac{(\alpha^2 + 2\alpha\beta) \beta^2}{2} - \alpha^2\beta^2 - \left( \frac{\alpha^4}{4} - \frac{(2\alpha + \beta) \alpha^3}{3} + \frac{(\alpha^2 + 2\alpha\beta) \alpha^2}{2} - \alpha^2\beta \alpha \right) \right)
    \]

11. After simplification, we get:
    \[
    = \frac{|a|}{12} (\beta - \alpha)^4
    \]

The final answer is \(\boxed{\frac{|a|}{12} (\beta - \alpha)^4}\).