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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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