Volume: Using Disks and Washers

AP Calculus AB

Volume: Using Disksand Washers

Jeff BlakeMatt Braddock

Elisabeth Mueller

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

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.

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

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

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

Presenting Volume

Similarly, you can cut a cantaloupe in half and remove the pulp, then you can identify cross-sections that are washers.

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)

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

2010 Exam Form B #1

Find the volume of the solid generated when R is revolved about the horizontal line y = 6.

2014 Exam #2

Find the volume of the solid generated when R is rotated about the horizontal line y = -2.

2008 Exam Form B #1

Find the volume of the solid generated when R is rotated about the vertical line x = -1.

2008 Exam Form B #1

Find the volume of the solid generated when R is rotated about the vertical line x = -1.

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