Let R be the region bounded by the parabola y=x2 and the line y=4 .What is the volume of the solid generated when R is rotated about the line y=4 ?What is the volume of the solid generated when R is rotated about the line y=5 ? don't explan

### Volume when rotated about the line \( y = 4 \):
\[
V_1 = \pi \int_{-2}^{2} \left[(4 - x^2)^2 - 0^2\right] \, dx = \frac{256\pi}{15}
\]

### Volume when rotated about the line \( y = 5 \):
\[
V_2 = \pi \int_{-2}^{2} \left[(5 - x^2)^2 - 1^2\right] \, dx = \frac{704\pi}{15}
\]

Would you like any further details or clarification?

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Here are 5 related questions:
1. How does the choice of axis of rotation affect the volume?
2. What if the region were rotated about the x-axis instead?
3. How do you find the surface area of the solid generated by rotation?
4. What happens if the region is rotated about the line \( y = 3 \)?
5. How would the volume change if the parabola were \( y = x^2 + 1 \) instead?

**Tip:** Always ensure that your axis of rotation is properly accounted for when setting up the integral, as it can dramatically affect the calculation.

Calculate the volume of the solid formed by rotating the region bounded by the parabola y=x^2 and the line y=4 about the line y=4. Includes step-by-step integration process and formulas for cylindrical shells. Suitable for advanced high school students and calculus enthusiasts.

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