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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.

The graph of y = 4x^3 - 9x^2 is shown. Which of the following gives the area of the shaded section? What is the shaded area?

Learn how to calculate the area of the shaded region under the curve y = 4x^3 - 9x^2 between x = 0 and x = 1 using definite integration. This calculus problem covers areas under curves and applying the Fundamental Theorem of Calculus.

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