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The given integral is:

\[
\int_{0}^{25} \pi \left(5 - \frac{y}{5}\right)^2 \, dy
\]

Let's break this down step-by-step to solve it.

### Step 1: Simplify the integrand
First, expand the expression inside the integral:

\[
\left(5 - \frac{y}{5}\right)^2 = 25 - 2 \cdot 5 \cdot \frac{y}{5} + \left(\frac{y}{5}\right)^2
\]
\[
= 25 - 2y + \frac{y^2}{25}
\]

So, the integral becomes:

\[
\int_{0}^{25} \pi \left(25 - 2y + \frac{y^2}{25}\right) \, dy
\]

### Step 2: Separate the terms
We can now separate the integral into three simpler integrals:

\[
\pi \int_{0}^{25} \left(25 - 2y + \frac{y^2}{25}\right) \, dy = \pi \left( \int_{0}^{25} 25 \, dy - 2 \int_{0}^{25} y \, dy + \frac{1}{25} \int_{0}^{25} y^2 \, dy \right)
\]

### Step 3: Evaluate each integral

1. \(\int_{0}^{25} 25 \, dy = 25y \Big|_0^{25} = 25 \times 25 = 625\)
2. \(\int_{0}^{25} y \, dy = \frac{y^2}{2} \Big|_0^{25} = \frac{25^2}{2} = \frac{625}{2}\)
3. \(\int_{0}^{25} y^2 \, dy = \frac{y^3}{3} \Big|_0^{25} = \frac{25^3}{3} = \frac{15625}{3}\)

### Step 4: Substitute the results back
Now, substitute these results back into the original expression:

\[
\pi \left( 625 - 2 \times \frac{625}{2} + \frac{1}{25} \times \frac{15625}{3} \right)
\]

Simplifying each term:

- \(625\)
- \(- 625\)
- \(\frac{1}{25} \times \frac{15625}{3} = \frac{625}{3}\)

So the expression simplifies to:

\[
\pi \left( 625 - 625 + \frac{625}{3} \right) = \pi \times \frac{625}{3}
\]

### Final Answer:
\[
\frac{625\pi}{3}
\]

This is the value of the integral.

Would you like further details or explanations? 

Here are five follow-up questions for you:
1. How do you handle integrals involving squares in more complex cases?
2. Can you explain how to simplify a function like \( \left(5 - \frac{y}{5}\right)^2 \)?
3. What are other types of definite integrals that follow a similar structure?
4. How does the presence of constants outside the integral affect the solution process?
5. What are common techniques to solve polynomial integrals?

**Tip**: When evaluating definite integrals, always simplify the expression inside the integral before proceeding to make the computations easier.

Evaluate the definite integral \( \int_{0}^{25} \pi \left( 5 - \frac{y}{5} \right)^2 \, dy \).

Learn how to solve the definite integral \( \int_{0}^{25} \pi \left( 5 - \frac{y}{5} \right)^2 \, dy \) step-by-step. This problem covers key calculus concepts, including polynomial expansion and the evaluation of definite integrals. Suitable for undergraduate or advanced high school students.

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