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The integral you have provided is:

\[
\int_{-5}^{5} \left( \pi \left( \left( 7 + \sqrt{25 - y^2} \right)^2 - \left( 7 - \sqrt{25 - y^2} \right)^2 \right) \right) dy
\]

Let me evaluate this integral and provide the result.The value of the given integral is:

\[
350\pi^2
\]

Let me know if you would like further explanation on the steps I took to solve this!

Here are 5 related questions to explore further:
1. How does symmetry simplify the evaluation of definite integrals?
2. Can you expand on how the difference of squares affects integrals?
3. What happens to the integral if we change the limits of integration?
4. How do you interpret this result geometrically or physically?
5. Could this integral represent an area or volume in a real-world scenario?

**Tip:** When dealing with complex expressions inside an integral, checking for symmetry can often simplify the problem and speed up the calculation.

Evaluate the definite integral: \[ \int_{-5}^{5} \left( \pi \left( \left( 7 + \sqrt{25 - y^2} \right)^2 - \left( 7 - \sqrt{25 - y^2} \right)^2 \right) \right) dy \]

Learn how to evaluate a definite integral involving complex expressions with symmetry. This problem explores calculus concepts such as definite integrals, geometry of curves, and the application of the difference of squares. Suitable for advanced high school and college students.

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