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The problem provides a probability density function (PDF) for the particle size distribution \( f(x) \), where:

\[
f(x) =
\begin{cases}
3x^{-4}, & x > 1 \\
0, & \text{otherwise}.
\end{cases}
\]

We are tasked with finding the probability that a random particle has a size greater than 5 micrometers. This probability can be calculated using the cumulative distribution function (CDF) by finding the area under the PDF from \( x = 5 \) to infinity.

The probability that a particle exceeds 5 micrometers is:

\[
P(X > 5) = \int_5^\infty f(x) \, dx = \int_5^\infty 3x^{-4} \, dx.
\]

### Step 1: Calculate the integral

\[
\int_5^\infty 3x^{-4} \, dx = 3 \int_5^\infty x^{-4} \, dx.
\]

Now, integrate \( x^{-4} \):

\[
\int x^{-4} \, dx = \frac{x^{-3}}{-3}.
\]

So,

\[
3 \int_5^\infty x^{-4} \, dx = 3 \left[ \frac{x^{-3}}{-3} \right]_5^\infty.
\]

### Step 2: Evaluate the limits

For the upper limit as \( x \to \infty \), \( x^{-3} \to 0 \). Thus, the evaluation becomes:

\[
3 \left[ \frac{0}{-3} - \frac{5^{-3}}{-3} \right] = 3 \left( 0 + \frac{1}{3 \cdot 5^3} \right).
\]

\[
3 \cdot \frac{1}{375} = \frac{3}{375} = \frac{1}{125}.
\]

### Final Answer:
The probability that a random particle exceeds 5 micrometers is \( \frac{1}{125} \).

Would you like more details on the steps or need further clarification? Here are some related questions:

1. How do we interpret a probability density function in the context of particle sizes?
2. What would be the probability of a particle size exceeding 10 micrometers?
3. How does the integral of the PDF over its entire range equal 1?
4. Could this type of distribution apply to other physical phenomena? 
5. How would the probability change if the PDF were different, say \( 4x^{-5} \)?

**Tip:** When dealing with probability distributions, always check if the function integrates to 1 over the valid range to ensure it is a valid PDF.

What is the probability that a random particle from the manufactured fuel exceeds 5 micrometers?

Learn how to solve a probability density function problem where the particle size exceeds 5 micrometers. The solution involves integrating the given PDF and applying the Fundamental Theorem of Calculus. Ideal for undergraduate students.

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