solve the definite integral:

[\int_{2}^{4}(-x^2+6x-8)dx]

To solve the definite integral:

\[
\int_{2}^{4}(-x^2 + 6x - 8) \, dx
\]

we will follow these steps:

### Step 1: Find the indefinite integral
First, integrate the function term by term:

\[
\int (-x^2 + 6x - 8) \, dx
\]

- For \(-x^2\), the integral is:
  \[
  \int -x^2 \, dx = -\frac{x^3}{3}
  \]

- For \(6x\), the integral is:
  \[
  \int 6x \, dx = 3x^2
  \]

- For \(-8\), the integral is:
  \[
  \int -8 \, dx = -8x
  \]

So, the indefinite integral is:
\[
-\frac{x^3}{3} + 3x^2 - 8x + C
\]

### Step 2: Apply the limits of integration
Now, apply the limits of integration from \(x = 2\) to \(x = 4\):

\[
\int_{2}^{4} (-x^2 + 6x - 8) \, dx = \left[ -\frac{x^3}{3} + 3x^2 - 8x \right]_{2}^{4}
\]

This means we need to evaluate the expression at \(x = 4\) and \(x = 2\) and subtract:

1. At \(x = 4\):
   \[
   -\frac{(4)^3}{3} + 3(4)^2 - 8(4) = -\frac{64}{3} + 3(16) - 32 = -\frac{64}{3} + 48 - 32 = -\frac{64}{3} + 16
   \]
   \[
   = \frac{-64 + 48}{3} = \frac{-16}{3}
   \]

2. At \(x = 2\):
   \[
   -\frac{(2)^3}{3} + 3(2)^2 - 8(2) = -\frac{8}{3} + 3(4) - 16 = -\frac{8}{3} + 12 - 16 = -\frac{8}{3} - 4
   \]
   \[
   = \frac{-8 - 12}{3} = \frac{-20}{3}
   \]

### Step 3: Subtract the results
Now subtract the result at \(x = 2\) from the result at \(x = 4\):

\[
\left( \frac{-16}{3} \right) - \left( \frac{-20}{3} \right) = \frac{-16 + 20}{3} = \frac{4}{3}
\]

### Final Answer:
\[
\int_{2}^{4} (-x^2 + 6x - 8) \, dx = \frac{4}{3}
\]

Would you like a further breakdown or have any questions on this?

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Here are some related questions for deeper understanding:
1. How do you verify a definite integral using geometry?
2. What is the geometric interpretation of this integral on the coordinate plane?
3. How would you find the indefinite integral of a more complex polynomial?
4. How do you apply the Fundamental Theorem of Calculus to definite integrals?
5. Can this process be automated using numerical integration techniques?

**Tip:** Always double-check your integration by differentiating the result to ensure it matches the original function.

solve the definite integral: [\int_{2}^{4}(-x^2+6x-8)dx]

Learn how to solve the definite integral \(\int_{2}^{4}(-x^2+6x-8)dx\) with a step-by-step guide. This calculus problem covers the polynomial integration process, applying the Fundamental Theorem of Calculus, and is ideal for Grades 11-12 and early college students.

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