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5/23/17
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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
32:55
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
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