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Applications of Integration Volumes of Revolution Many thanks to http://mathdemos.gcsu.edu/shellmeth od/gallery/gallery.html

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Method of discs

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Take this ordinary line 2 5 Revolve this line around the x axis We form a cylinder of volume

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We could find the volume by finding the volume of small disc sections 2 5

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If we stack all these slices… We can sum all the volumes to get the total volume

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To find the volume of a cucumber… we could slice the cucumber into discs and find the volume of each disc.

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The volume of one section: Volume of one slice =

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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

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The volume of one section: r = y value h = dx Volume of one slice =

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Volume of cucumber… Area of 1 slice Thickness of slice

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Take this function… and revolve it around the x axis

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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=

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Take this shape…

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Revolve it…

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Christmas bell…

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Divide the region into strips

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Form a cylindrical slice

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Repeat the procedure for each strip

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To generate this solid

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A polynomial

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Regions that can be revolved using disc method

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Regions that cannot….

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Model this muffin.

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Washer Method

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A different cake

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Slicing….

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Making a washer

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Revolving around the x axis

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Region bounded between y = 1, x = 0, y = 1 x = 0

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Volume generated between two curves y= 1

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Area of cross section.. f(x) g(x)

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dx

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Your turn: Region bounded between x = 0, y = x,

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Region bounded between y =1, x = 1

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Region bounded between

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Around the x axis- set it up

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Revolving shapes around the y axis

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Region bounded between

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Volume of one washer is

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Calculate the volume of one washer

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And again…region bounded between y=sin(x), y = 0.

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Region bounded between x = 0, y = 0, x = 1,

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Worksheet 5 Delta Exercise 16.5

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Volumes by Slicing. disk Find the Volume of revolution using the disk method washer Find the volume of revolution using the washer method shell Find the.

Volumes by Slicing. disk Find the Volume of revolution using the disk method washer Find the volume of revolution using the washer method shell Find the.

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