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Applications of Integration Volumes of Revolution Many thanks to http://mathdemos.gcsu.edu/shellmeth od/gallery/gallery.html
Method of discs
Take this ordinary line 2 5 Revolve this line around the x axis We form a cylinder of volume
We could find the volume by finding the volume of small disc sections 2 5
If we stack all these slices… We can sum all the volumes to get the total volume
To find the volume of a cucumber… we could slice the cucumber into discs and find the volume of each disc.
The volume of one section: Volume of one slice =
We could model the cucumber with a mathematical curve and revolve this curve around the x axis… Each slice would have a thickness dx and height y. 25 -5
The volume of one section: r = y value h = dx Volume of one slice =
Volume of cucumber… Area of 1 slice Thickness of slice
Take this function… and revolve it around the x axis
We can slice it up, find the volume of each disc and sum the discs to find the volume….. Radius = y Area = Thickness of slice = dx Volume of one slice=
Take this shape…
Divide the region into strips
Form a cylindrical slice
Repeat the procedure for each strip
To generate this solid
Regions that can be revolved using disc method
Regions that cannot….
Model this muffin.
A different cake
Making a washer
Revolving around the x axis
Region bounded between y = 1, x = 0, y = 1 x = 0
Volume generated between two curves y= 1
Area of cross section.. f(x) g(x)
Your turn: Region bounded between x = 0, y = x,
Region bounded between y =1, x = 1
Region bounded between
Around the x axis- set it up
Revolving shapes around the y axis
Region bounded between
Volume of one washer is
Calculate the volume of one washer
And again…region bounded between y=sin(x), y = 0.
Region bounded between x = 0, y = 0, x = 1,
Worksheet 5 Delta Exercise 16.5
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The region enclosed by the x-axis and the parabola is revolved about the line x = –1 to generate the shape of a cake. What is the volume of the cake? DO.
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6.3 Volumes of Revolution Tues Dec 15 Do Now Find the volume of the solid whose base is the region enclosed by y = x^2 and y = 3, and whose cross sections.
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Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, Disk and Washer Methods Limerick Nuclear Generating Station, Pottstown,
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Lesson 6-2c Volumes Using Washers. Ice Breaker Volume = ∫ π(15 - 8x² + x 4 ) dx x = 0 x = √3 = π ∫ (15 - 8x² + x 4 ) dx = π (15x – (8/3)x 3 + (1/5)x 5.
7.3 VOLUMES. Solids with Known Cross Sections If A(x) is the area of a cross section of a solid and A(x) is continuous on [a, b], then the volume of the.
Outer radius inner radius thickness of slice cylinder.
Volume: The Disc Method Section 6.2. If a region in the plane is revolved about a line, the resulting solid is a solid of revolution, and the line is.
A k = area of k th rectangle, f(c k ) – g(c k ) = height, x k = width. 6.1 Area between two curves.
DO NOW: Find the volume of the solid generated when the region in the first quadrant bounded by the given curve and line is revolved about the x-axis.
Adapted by Mrs. King from
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