Volume Calculation: Disc and Washer Methods

Overview

• Disc Method: Used for calculating volume by rotating a region around an axis (x-axis or y-axis). Involves creating discs or cylinders.
• Washer Method: Similar to the disc method but used when the region involves a hole in the middle, like a washer.

Disc Method

Around the X-Axis

1. Curve: Given a curve between points a and b.
2. Cross-Sectional Area: Consider a small disc or cylinder with radius r and height dx.
3. Volume of Cylinder (Disc): V = \pi r^2 h
4. Integral: Sum the volumes of all discs from a to b. V = \pi \int_{a}^{b} r^2(x) dx
5. Example: Function y = \sqrt{x}, bounds x = 0 to x = 4.
   • Radius: r(x) = \sqrt{x}
   • Integral: V = \pi \int_{0}^{4} (\sqrt{x})^2 dx V = \pi \int_{0}^{4} x dx V = \pi [\frac{x^2}{2}]_{0}^{4} V = 8\pi

Around the Y-Axis

1. Curve: Given a curve between points c and d.
2. Cross-Sectional Area: Similar process but consider the axis of rotation and change of variable.
3. Integral: Sum the volumes of all discs from c to d. V = \pi \int_{c}^{d} r^2(y) dy
4. Example: Function y = x^2, bounds y = 0 to y = 4.
   • Radius: r(y) = \sqrt{y} (since y = x^2)
   • Integral: V = \pi \int_{0}^{4} (\sqrt{y})^2 dy V = \pi \int_{0}^{4} y dy V = \pi [\frac{y^2}{2}]_{0}^{4} V = 8\pi

Washer Method

Example: Rotating around the X-Axis

1. Function: y = \frac{1}{x}, bounds x = 1 to x = 3.
2. Radius: Distance between the curve and axis of rotation, r(x) = \frac{1}{x}.
3. Integral: V = \pi \int_{1}^{3} (\frac{1}{x})^2 dx V = \pi \int_{1}^{3} x^{-2} dx V = \pi [-\frac{1}{x}]_{1}^{3} V = \pi [-\frac{1}{3} - (-1)] V = \frac{2\pi}{3}

Example: Rotating around the Y-Axis

1. Function: y = x^{\frac{2}{3}}, bounds y = 0 to y = 1.
2. Radius: r(y) = y^{\frac{3}{2}}
3. Integral: V = \pi \int_{0}^{1} (y^{\frac{3}{2}})^2 dy V = \pi \int_{0}^{1} y^3 dy V = \pi [\frac{y^4}{4}]_{0}^{1} V = \frac{\pi}{4}

Key Points

• For the disc method, always ensure you are using the correct variable for integration (dx for x-axis and dy for y-axis).
• Understand the geometry of the region to correctly identify the radius function.
• Integrate the area of the circular cross-section to get the volume.
• Washer method involves subtracting inner radius from outer radius if there's a hollow center.