Applications of integration
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Applications of Integration. Volumes of Revolution Many thanks to http:// mathdemos.gcsu.edu / shellmethod /gallery/ gallery.html. Method of discs. Take this ordinary line. Revolve this line around the x axis. 2. 5. We form a cylinder of volume.

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Applications of Integration

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Applications of Integration

Volumes of Revolution

Many thanks to

http://mathdemos.gcsu.edu/shellmethod/gallery/gallery.html


Method of discs


Take this ordinary line

Revolve this line around the x axis

2

5

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…

25

-5

Each slice would have a thickness dx and height y.


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…..

Volume of one slice=

Radius = y

Area =

Thickness of slice = dx


Take this shape…


Revolve it…


Christmas bell…


Divide the region into strips


Form a cylindrical slice


Repeat the procedure for each strip


To generate this solid


A polynomial


Regions that can be revolved using disc method


Regions that cannot….


Model this muffin.


Washer Method


A different cake


Slicing….


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)


dx


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


Region bounded between y =1, x = 1


Region bounded betweeny = 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 betweeny=sin(x), y = 0.


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


Worksheet 5Delta Exercise 16.5


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