📘 Volume from Rotation

🔹 A) Around Horizontal Axis (x-axis)

Disk Method (no hole)

$$V = \pi \int_a^b [R(x)]^2\,dx$$

Washer Method (with hole)

$$V = \pi \int_a^b \big( [R(x)]^2 - [r(x)]^2 \big)\,dx$$

• $R(x)$ = outer radius
• $r(x)$ = inner radius

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🔹 Example (Disk)

Rotate $y = x$ from $0$ to $1$ around x-axis:

$$V = \pi \int_0^1 x^2\,dx = \pi \left[\frac{x^3}{3}\right]_0^1 = \frac{\pi}{3}$$

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🔹 B) Around Vertical Axis (y-axis)

Shell Method

$$V = 2\pi \int_a^b (\text{radius})(\text{height})\,dx$$

• radius = distance to axis
• height = function value

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🔹 Example (Shell)

Rotate $y = x$ from $0$ to $1$ around y-axis:

$$V = 2\pi \int_0^1 x \cdot x \,dx = 2\pi \int_0^1 x^2\,dx = 2\pi \cdot \frac{1}{3} = \frac{2\pi}{3}$$

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🔹 Summary Table

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🔹 Key Takeaways

• Disk/Washer → slices ⟂ axis
• Shell → slices ∥ axis
• Always define radius clearly
• Sketch helps avoid mistakes