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Volumes of Revolution By Shayna Herns, Alyssa Ioannou, and Tony Jongco

Volumes of Revolution Basic Geometric Concept 1.  Take an enclosed area on a plane 2.  Place a line on the same plane: that will be the axis of

rotation 3.  Rotate the enclosed curve around the line for one

revolution 4.  The final result is 3-D shape with a hole in the middle

How is the shape created? 1.  The shape on the plane is created

by first finding the area between two functions from an interval a to b

2.  The most common axis of rotation used is about the x-axis, but any line be used

3.  The area from step one is rotated about the axis mentioned in step two

4.  The final result is a volume of revolution

Using Area to Find Volume ●  The outer function is my outer radius, and the inside function is my inner

radius. ●  I know the formula for the area of a circle

○  pi*r^2. ●  So the area of the washer is pi*(outer radius)^2-pi*(inner radius)^2 . ●  Plugging the functions into our respective radii will give us a formula for the

washer method we can use. ●  The outer function would replace the outer radius and the inner function would

replace the inner radius.

Why is the Washer Method used?

●  We approximate this area with 2-D rectangles; the smaller the width of these rectangles (∆x), the better we can approximate the area under the curve

●  Likewise in our 3-D version, the thinner the width of our “washers”, the better we can approximate the volume of our solid

●  One way to think of this method is as the 3-D version of finding the area under a curve

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Why is the Washer Method used? This is a tangible and precise method used to find the volume of a solid of revolution: ●  In order to find the volume of the solid, we

can imagine the interval from A to B divided up into many extremely thin intervals of width dx

●  This slices the solid into many extremely thin washers, whose exact shapes depend on the functions we began with

●  The volume of each washer is thus its area times its width, or A(x) dx

●  By adding up the volume of all of these washers, we can find the volume of the entire solid

Conceptual Blend

Conceptual Blend

Conceptual Blend

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Blended Space: The Washer Method Linguistic Evidence Supporting

Conceptual Blend Hypothesis Khan Academy Video “Generalizing the Washer Method” ●  1:30 - “imagine taking a chunk between these two functions … and

let’s rotate that whole thing around the x-axis. If we rotate this around the x-axis we end up with a washer”

●  2:02 - “it’s really just kind of the disc method where you’re gutting out the inside of a disk”

●  2:47 - “a washer you could imagine is kind of a gutted out coin” ●  8:23 - “we conceptualized it as a washer”

Khan Academy Video “Disc Method (Washer Method) for Rotation Around X-Axis”

●  6:24 - “you can imagine a quarter that has an infinitely thin depth”

Notational Evidence Supporting Conceptual Blend Hypothesis

Khan Academy Video “Generalizing the Washer Method” Jon Rogawski’s Calculus: Second Edition

Fictive Motion ●  A function does not actually move as it exists in a

specific location on the plane ●  That is, any given point on the function satisfies a

specific, immovable set of Cartesian coordinates ●  However, imagining the function as “moving” about an

axis allows us to imagine a volume of revolution in a Cartesian plane

●  Integral to the concept of rotating the functions about an axis is the ability to perceive static functions that exist on a 2-D plane as moving through 3-D space

Linguistic Evidence Supporting Fictive Motion Hypothesis

Jon Rogawski’s Calculus: Second Edition ●  P. 375 - “each of these [sphere and cone] is swept out as a plane

region revolves around the axis”

Khan Academy Video “Generalizing the Disc Method Around X-Axis”

●  0:52 - “We’re going between a and b. These are just two endpoints along the x-axis”

Khan Academy Video “Disk Method (Washer Method) for Rotation Around X-Axis” ●  0:54 - “we hollow out a cone inside of it” ●  1:46 - “we carve out a cone in the center”

Linguistic Evidence Supporting Fictive Motion Hypothesis

Khan Academy Video “Disc Method Around X-Axis” ●  3:13 - “the x-axis would pop the base right over there, would go right

through the base and then come out on the other side”

“Volumes of Solids of Revolution / Method of Rings” from Paul’s Online Notes

●  “One of the easier methods for getting the cross-sectional area is to cut the object perpendicular to the axis of rotation”

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Other Evidence Supporting Fictive Motion

ProfRobBob (Youtube) “Volumes of Solid of Revolution Disk Method and Washer Method” ●  Co-speech-gesture production ●  Notational evidence

Other Evidence Supporting Fictive Motion

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Final Conclusions ●  The Washer Method is a means of finding the volume of a solid of

revolution ●  Our understanding of this process is based on the cognitive mechanisms

of conceptual blending and fictive motion o  The conceptual blend takes the inputs of “volume of revolution” and

“stack of washers” to form the Washer Method o  Fictive motion allows us to imagine static functions as able to rotate

through three-dimensional space about an axis ●  These cognitive mechanisms are seen in textbooks, videos, and other

teaching tools o  Types of evidence include linguistic expressions, notational devices,

and co-speech gesture production