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To find the area of the shaded region between the curves \( f(x) = x^3 + x^2 - 6x \) and \( g(x) = 6x \), we need to set up the integral of the difference between these functions over the interval where they intersect.

### Steps:
1. **Find the points of intersection** of the two curves, \( f(x) = g(x) \). 
   Set \( x^3 + x^2 - 6x = 6x \), which simplifies to:
   \[
   x^3 + x^2 - 12x = 0
   \]
   Factor the equation:
   \[
   x(x^2 + x - 12) = 0
   \]
   Solve \( x^2 + x - 12 = 0 \) using the quadratic formula:
   \[
   x = \frac{-1 \pm \sqrt{1^2 - 4(1)(-12)}}{2(1)} = \frac{-1 \pm \sqrt{1 + 48}}{2} = \frac{-1 \pm \sqrt{49}}{2}
   \]
   This gives \( x = 3 \) and \( x = -4 \). Thus, the points of intersection are \( x = -4 \), \( x = 0 \), and \( x = 3 \).

2. **Set up the integral** to compute the area between the curves over the two regions:
   - From \( x = -4 \) to \( x = 0 \), the curve \( g(x) = 6x \) is above \( f(x) \).
   - From \( x = 0 \) to \( x = 3 \), the curve \( f(x) \) is above \( g(x) \).

   The total area is given by:
   \[
   A = \int_{-4}^{0} (g(x) - f(x)) \, dx + \int_{0}^{3} (f(x) - g(x)) \, dx
   \]

3. **Calculate each integral**:
   - First integral:
     \[
     \int_{-4}^{0} \left(6x - (x^3 + x^2 - 6x)\right) \, dx = \int_{-4}^{0} (-x^3 - x^2 + 12x) \, dx
     \]
   - Second integral:
     \[
     \int_{0}^{3} \left((x^3 + x^2 - 6x) - 6x\right) \, dx = \int_{0}^{3} (x^3 + x^2 - 12x) \, dx
     \]

I will now calculate these integrals to find the total area.The total area of the shaded region is \( \frac{937}{12} \). 

It appears the correct option wasn't selected in the original problem, as the area is not \( \frac{81}{12} \) but rather \( \frac{937}{12} \).

Would you like further details on the calculation or clarification of any steps?

Here are five related questions to deepen your understanding:

1. How do you determine which curve is above the other in a region between two curves?
2. Why is it important to split the integral based on points of intersection?
3. How would the area change if the functions were shifted up or down?
4. Can you generalize the process of finding the area between two curves for any polynomial functions?
5. What happens to the integral if the curves do not intersect within the interval?

**Tip:** When setting up integrals to find the area between curves, always check which function is greater on each interval, as this determines the order of subtraction in the integral.

Find the area of the shaded region. f(x) = x^3 + x^2 - 6x, g(x) = 6x

Learn how to find the area between the curves f(x) = x^3 + x^2 - 6x and g(x) = 6x. This step-by-step solution covers integral calculus concepts, the quadratic formula for intersection points, and applying the fundamental theorem of calculus.

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