volume: using disks and washers
DESCRIPTION
Overview of volume using disks and washers (for instructors)TRANSCRIPT
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AP Calculus AB
Volume: Using Disksand Washers
Jeff BlakeMatt Braddock
Elisabeth Mueller
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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
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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.
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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
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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
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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
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Presenting Volume
Similarly, you can cut a cantaloupe in half and remove the pulp, then you can identify cross-sections that are washers.
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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)
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AP Free Response Questions1998 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
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2010 Exam Form B #1
Find the volume of the solid generated when R is revolved about the horizontal line y = 6.
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2014 Exam #2
Find the volume of the solid generated when R is rotated about the horizontal line y = -2.
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2008 Exam Form B #1
Find the volume of the solid generated when R is rotated about the vertical line x = -1.
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2008 Exam Form B #1
Find the volume of the solid generated when R is rotated about the vertical line x = -1.
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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