Volume: Using Disks and Washers
- 1. AP Calculus AB Volume: Using Disks and Washers Jeff Blake Matt Braddock Elisabeth Mueller
- 2. Volume Overview Area between curves can be rotated about horizontal or vertical lines to create solids with circular cross-sections ● The cross-sections of the solid are disks ● If there is a hole in the solid, the cross-sections are washers
- 3. Volume Overview Approximates the volume of a solid by summing the volumes of the cross-sections with width Δx, then uses the limit to shrink the widths to zero. Equations ● Disk Method: R(x) is the radius of the disk ● Washer Method: R(x) is the outside radius of the washer, and r(x) is the inside radius of the washer.
- 4. Essential Understanding ● Find the intersection(s) of the graphs and use them as limits of integration ● Find the area of the region between the graph and the axis, or between two graphs ● Determine when to use the disk method and when to use the washer method ● Determine the outside and inside radii of a washer ● Determine when to integrate with respect to x and when to integrate with respect to y
- 5. Issues with Content ● Determining which variable to integrate with respect to ● Determining the radius/radii of the cross-section (especially when the axis of revolution is not the x- or y-axis) ● Determining limits of integration when functions intersect
- 6. Presenting Volume ● A cucumber is approximately a cylinder (removing the ends makes this more apparent) ● Throughout the cucumber, the radius is inconsistent, so the volume formula does not apply ● Can cut the cucumber into disks with a consistent radius, determine the volume of each disk, then sum the volumes
- 7. Presenting Volume Similarly, you can cut a cantaloupe in half and remove the pulp, then you can identify cross-sections that are washers.
- 8. Remediation Remediation Videos ● Finding zeros and intersects with the TI-84 (video) ● Finding area underneath a curve (video) ● Evaluating integrals analytically (video) Video Lessons for Volume ● Educreations – Introduction (video) – Additional problems (video) ● Khan Academy – Set of lessons (video)
- 9. AP Free Response Questions 1998 AB1 2006 AB1 1999 AB2 2006B AB1 2000 AB1 2007 AB1 2001 AB1 2007B AB1 2002 AB1 2008B AB1 2002B AB1 2009 AB4* 2003 AB1 2009B AB4* 2003B AB1 2010 AB4* 2004 AB2 2010B AB1 2004B AB1 2011 AB3* 2005 AB1 2013 AB5* 2005B AB1 2014 AB2 Volume of a solid of revolution ● Appears on nearly every AP Calculus AB exam ● Until recently, always calculator active ● 24 free response questions available on AP Central *Graphing Calculator Non-Active
- 10. 2010 Exam Form B #1 Find the volume of the solid generated when R is revolved about the horizontal line y = 6.
- 11. 2014 Exam #2 Find the volume of the solid generated when R is rotated about the horizontal line y = -2.
- 12. 2008 Exam Form B #1 Find the volume of the solid generated when R is rotated about the vertical line x = -1.
- 13. 2008 Exam Form B #1 Find the volume of the solid generated when R is rotated about the vertical line x = -1.
- 14. Opportunities for Technology ● Calculators are used to find intersections of graphs and compute volume integrals ● Visual representations of creating solids available online (disk method; washer method) ● Calculus in Motion for Geometer's Sketchpad