We are given the function $y = 4x^3 - 9x^2$, and we are asked to find the area of the shaded region from $x = 0$ to $x = 1$ based on the graph.

Steps to Solve:

1. Determine the area under the curve: The area of the shaded region can be calculated using definite integration. We need to integrate the given function from $x = 0$ to $x = 1$.

   $\text{Area} = \int_0^1 (4x^3 - 9x^2) \, dx$

2. Compute the integral: $\int (4x^3 - 9x^2) \, dx = x^4 - 3x^3 + C$ Now, we apply the limits of integration from 0 to 1.

3. Evaluate the definite integral: Substituting the limits $x = 1$ and $x = 0$ into the antiderivative:

   $\left[ x^4 - 3x^3 \right]_0^1 = \left( 1^4 - 3(1^3) \right) - \left( 0^4 - 3(0^3) \right)$ Simplifying this: $(1 - 3) - (0 - 0) = -2$

4. Area interpretation: Since the integral gives $-2$, and area cannot be negative, the absolute value is taken. Thus, the area of the shaded region is:

   $\text{Shaded Area} = 2$

Correct Option:

From the given options, the correct expression for the area is: $\boxed{\text{Option C: } \left[ x^4 - 3x^3 \right]_0^1}$

Would you like further details or clarifications? Here are some related questions to expand on the concept:

1. How do you handle negative areas in definite integrals?
2. What does the graph of $4x^3 - 9x^2$ tell us about its roots and turning points?
3. Can you calculate the area between the curve and the x-axis when the curve is below the axis?
4. How do you set up a definite integral for more complex curves?
5. How does the Fundamental Theorem of Calculus connect differentiation and integration?

Tip: When integrating to find areas, always check the sign of the result. If the area is negative, this indicates the region is below the x-axis, so take the absolute value to find the true area.