Calculus Lecture: Volumes of a Solid of Revolution - Washer Method

Introduction

• Focus on volumes of a solid of revolution, specifically using the washer method.
• Washer method:
  • Used when cross-section is a circle with a gap in the middle.
  • Formula: ( \pi (R^2 - r^2) ), where ( R ) is the big radius and ( r ) is the small radius.
• Previously revolved around x or y axis, now revolving around a new line.

Revolving Around New Lines

• Example 1: Revolve around the line ( y = 2 ).
  • Involves creating a mirror image and revolving the shape.
  • Visualizing the shape: Big shape with a hole in the middle.

Setting Up Equations

1. Conventional Method (Textbook):

   • Integral from 0 to 1, ( \pi ) at the front.
   • Use ( R^2 - r^2 ) with respect to x.
   • Big radius: ( 2 - x^2 ).
   • Small radius: ( 2 - \sqrt{x} ).
   • Remember ( \pi ) for area of circles.

2. Alternative Method (Easier):

   • Shift line down 2 units, equations change.
   • New equations: ( y = \sqrt{x} - 2 ), ( y = x^2 - 2 ).
   • Treat shifted line as x-axis.
   • Big radius from x-axis: ( x^2 - 2 ).
   • Small radius from x-axis: ( \sqrt{x} - 2 ).
   • Resulting integrals are equivalent due to squaring property.

Additional Example: Vertical Line

• Example 2: Revolve around line ( x = -1 ).
  • Requires equations in terms of y.
  • Solve equations for y and find mirror image.
  • Revolve around the vertical line.

Setting Up Equations

1. Standard Method:

   • Integrate ( R^2 - r^2 ) from y=1 to y=5.
   • Big radius: ( 1 + \sqrt{y - 1} ).
   • Small radius: ( 1 + \frac{y-1}{2} ).

2. Alternative Method:

   • Shift functions one unit right.
   • New equations with +1 shift.
   • Equivalent setup with respect to the new y-axis.

Preparing for the AP Exam

• Practice Exams:
  • Take as many as possible, time yourself.
  • Correct errors to understand mistakes.
• Resources:
  • Barron's AP Review or Princeton Review books.
• Stress Management:
  • Focus on getting around 65% right for a potential 5 score.
  • Don’t stress over hard questions.
• Procedural Skills:
  • Automatic understanding of derivatives and integrals.
  • Know chain rule and u-substitution.

Conclusion

• Completion of class and preparation for exams.
• AB and BC students have specific advice for final preparations.
• Signing off message from Mr. Bean.