Calculating Volume by Rotation

Overview

• Discussion on methods to calculate volume by rotating a region around the x-axis or y-axis.
• Methods include the Disk Method and the Washer Method.

Disk Method

• Concept: Involves revolving a region around an axis to form a disk shape.
• Volume of a Cylinder: Calculated using the formula ( V = \pi r^2 h ) where ( r ) is the radius and ( h ) is the height.
• Cross-sections: Typically circles with area ( \pi r^2 ).
• Integration: To find the volume, integrate ( \pi r^2 ) over the interval from A to B.

Rotating Around the X-axis

• Variables: In terms of ( x ).
• Integration formula: [ V = \pi \int_{a}^{b} r^2(x) , dx ]

Rotating Around the Y-axis

• Variables: In terms of ( y ).
• Integration formula: [ V = \pi \int_{c}^{d} r^2(y) , dy ]

Examples

Example 1: Function ( y = \sqrt{x} ) Rotated About X-axis

• Boundaries: From ( x = 0 ) to ( x = 4 ).
• Radius: ( r(x) = \sqrt{x} ).
• Integration: [ V = \pi \int_{0}^{4} (\sqrt{x})^2 , dx = \pi \int_{0}^{4} x , dx ]
• Solution: ( V = 8\pi ).

Example 2: Function ( y = \frac{1}{x} ) Rotated About X-axis

• Boundaries: From ( x = 1 ) to ( x = 3 ).
• Radius: ( r(x) = \frac{1}{x} ).
• Integration: [ V = \pi \int_{1}^{3} (\frac{1}{x})^2 , dx = \pi \int_{1}^{3} x^{-2} , dx ]
• Solution: ( V = \frac{2\pi}{3} ).

Example 3: Function ( y = x^2 ) Rotated About Y-axis

• Boundaries: From ( x = 0 ) to ( y = 4 ).
• Radius: ( r(y) = \sqrt{y} ).
• Integration: [ V = \pi \int_{0}^{4} y , dy ]
• Solution: ( V = 8\pi ).

Example 4: Function ( y = x^{2/3} ) Rotated About Y-axis

• Boundaries: From ( x = 0 ) to ( y = 1 ).
• Radius: ( x = y^{3/2} ).
• Integration: [ V = \pi \int_{0}^{1} y^3 , dy ]
• Solution: ( V = \frac{\pi}{4} ).

Conclusion

• Methods provide a systematic way to calculate the volume of solids formed by rotation around different axes.
• Essential to define the radius in terms of the axis variable and integrate accordingly.