To find the area under the curve $y = x^2(4 - x)$ between the specified limits, we'll use definite integration. The area is calculated by integrating the function $y = x^2(4 - x)$ with respect to $x$ over the interval from $x = 0$ to $x = 4$.

Step 1: Set up the integral

The function given is: $y = x^2(4 - x)$

Expanding the function: $y = 4x^2 - x^3$

The area under the curve from $x = 0$ to $x = 4$ is given by: $\text{Area} = \int_{0}^{4} (4x^2 - x^3) \, dx$

Step 2: Compute the integral

We will integrate each term separately: $\int (4x^2 - x^3) \, dx = \int 4x^2 \, dx - \int x^3 \, dx$

The integrals of each term are: $\int 4x^2 \, dx = \frac{4x^3}{3}$ $\int x^3 \, dx = \frac{x^4}{4}$

So, the integral becomes: $\text{Area} = \left[\frac{4x^3}{3} - \frac{x^4}{4}\right]_{0}^{4}$

Step 3: Evaluate the definite integral

Substitute the limits $x = 4$ and $x = 0$ into the expression: [ \text{Area} = \left[\frac{4(4)^3}{3} - \frac{(4)^4}{4}\right] - \left[\frac{4(0)^3}{3} - \frac{(0)^4}{4}\right] ]

[ = \left[\frac{4 \cdot 64}{3} - \frac{256}{4}\right] - \left[0 - 0\right] ]

$= \left[\frac{256}{3} - 64\right]$

$= \left[\frac{256 - 192}{3}\right] = \frac{64}{3} \approx 21.33$

So, the area under the curve is approximately 21.33 square units.

Do you want more details or have any questions?

Here are some related questions for further practice:

1. What is the area under the curve from $x = 1$ to $x = 3$?
2. How would you interpret the area in the context of real-world problems?
3. What does the negative area imply in a definite integral?
4. How do you handle curves that cross the x-axis when calculating the area?
5. How does the area change if the limits of integration are shifted?
6. What is the geometric significance of the result you obtained?
7. Can you find the x-coordinates where the curve intersects the x-axis?
8. What role does symmetry play in calculating the area under a curve?

Tip: When computing areas under curves, ensure that the function is properly integrated and evaluate the limits carefully to avoid sign errors.