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Understanding the Washer Method in Calculus

May 10, 2025

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.
    • Donโ€™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.