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The problem involves finding the area of the shaded region under the curve of the function \( f(x) = x^2 - 4x \) over the interval \([0, 8]\). 

### Solution:

To find the area, we need to calculate the definite integral of the function from \( x = 0 \) to \( x = 8 \).

The integral to solve is:

\[
\text{Area} = \int_{0}^{8} (x^2 - 4x) \, dx
\]

Let's compute this integral step by step.

1. **Find the antiderivative of \(f(x)\):**

   The antiderivative of \( x^2 \) is \(\frac{x^3}{3}\), and the antiderivative of \(-4x\) is \(-2x^2\). Therefore, the antiderivative of \(f(x)\) is:

   \[
   F(x) = \frac{x^3}{3} - 2x^2 + C
   \]

2. **Evaluate the definite integral from 0 to 8:**

   Substitute the limits into the antiderivative:

   \[
   F(8) = \frac{8^3}{3} - 2(8^2) = \frac{512}{3} - 128
   \]

   \[
   F(0) = \frac{0^3}{3} - 2(0^2) = 0
   \]

   Therefore, the area is:

   \[
   \text{Area} = \left(\frac{512}{3} - 128\right) - 0 = \frac{512}{3} - 128
   \]

3. **Simplify the expression:**

   Convert \(128\) to a fraction with the same denominator:

   \[
   128 = \frac{384}{3}
   \]

   So, the area becomes:

   \[
   \text{Area} = \frac{512}{3} - \frac{384}{3} = \frac{128}{3}
   \]

4. **Final Answer:**

   \[
   \text{Area} = \frac{128}{3} \approx 42.67
   \]

### Would you like more details or have any questions?

Here are some related questions to explore:

1. How do we determine the limits of integration for finding an area under a curve?
2. What does the definite integral represent in terms of area?
3. How would the calculation change if the interval was different, e.g., \([0, 5]\)?
4. What are some applications of finding the area under a curve in real-life situations?
5. How can this problem be solved graphically using software tools?

**Tip:** Always ensure the function remains above or below the x-axis consistently within the interval when calculating the area directly using integrals!

Given that f(x) = x^2 - 4x and x is in the real interval [0, 8], find the area of the shaded region.

Learn how to calculate the area under the curve for the function f(x) = x^2 - 4x in the interval [0, 8]. This problem involves applying calculus concepts such as definite integration and is suitable for high school students.

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