Volume Calculation using Dishwasher and Cylindrical Shell Methods

Overview

• The lecture covers two methods for finding the volume of a three-dimensional object: the Dishwasher Method and the Cylindrical Shell Method.
• Key difference: The shape of the object being analyzed dictates which method to use.

Dishwasher Method

Example Problem

• Function: ( y = x^3 )
• Goal: Find the volume of the area between the y-axis and the function, rotated around the y-axis.

Steps

1. Identify Area and Rotation

   • Identify the cross-section and determine the axis of rotation.
   • Integrate with respect to y if the cross-section is on the y-axis.

2. Define the Integral

   • Integral from ( a ) (bottom) to ( b ) (top): [ V = \int_a^b A(y) , dy ]
   • ( A(y) ): Area of the cross-section.

3. Calculate Area of Cross-section

   • For a circle: ( A(y) = \pi r^2 )
   • Radius ( r = x ), where ( x = y^{1/3} ).
   • Substitute: ( A(y) = \pi y^{2/3} )

4. Evaluate the Integral

   • ( V = \pi \int_0^8 y^{2/3} , dy )
   • Find the anti-derivative and evaluate: [ V = \pi \left. \frac{3}{5} y^{5/3} \right|_0^8 ]
   • Result: ( V = \frac{96}{5} \pi )_

Cylindrical Shell Method

Example Problem

• Function: ( y = 2x^2 - x^3 )
• Goal: Find the volume of the area between the function and the x-axis, rotated around the y-axis.

Steps

1. Identify Shape and Rotation

   • If the shape is not a half-cucumber, use the cylindrical shell method.
   • Integrate with respect to x if the base of the cylinder is on the x-axis.

2. Define the Integral

   • [ V = \int_a^b 2\pi \cdot r(x) \cdot h(x) , dx ]
   • ( r(x) = x ), ( h(x) = 2x^2 - x^3 ).

3. Evaluate the Integral

   • ( V = 2\pi \int_0^2 x(2x^2 - x^3) , dx )
   • Simplify and find the anti-derivative: [ V = 2\pi \left( \frac{1}{2}x^4 - \frac{1}{5}x^5 \right) \Bigg|_0^2 ]
   • Result: ( V = \frac{16}{5} \pi )_

Conclusion

• The volume of different shapes can be effectively calculated through appropriate integration methods.
• The choice between the dishwasher and cylindrical shell methods depends on the object's shape and axis of rotation.