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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

Applications of Integration

Volumes of Revolution

Many thanks to

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



Take this ordinary line
Take this ordinary line

Revolve this line around the x axis

2

5

We form a cylinder of volume



If we stack all these slices
If we stack all these slices… sections

We can sum all the volumes to get the total volume


To find the volume of a cucumber… sections

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


The volume of one section
The volume of one section: sections

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 section1
The volume of one section: revolve this curve around the

r = y value

h = dx

Volume of one slice =


Volume of cucumber
Volume of cucumber… revolve this curve around the

Area of 1 slice

Thickness of slice


Take this function
Take this function… revolve this curve around the

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
Take this shape… discs to find the volume…..


Revolve it
Revolve it… discs to find the volume…..


Christmas bell
Christmas bell… discs to find the volume…..


Divide the region into strips
Divide the region into strips discs to find the volume…..


Form a cylindrical slice
Form a cylindrical slice discs to find the volume…..


Repeat the procedure for each strip
Repeat the procedure for each strip discs to find the volume…..


To generate this solid
To generate this solid discs to find the volume…..


A polynomial
A polynomial discs to find the volume…..


Regions that can be revolved using disc method
Regions that can be revolved using disc method discs to find the volume…..


Regions that cannot
Regions that cannot…. discs to find the volume…..


Model this muffin
Model this muffin. discs to find the volume…..


Washer method
Washer Method discs to find the volume…..


A different cake
A different cake discs to find the volume…..


Slicing
Slicing…. discs to find the volume…..


Making a washer
Making a washer discs to find the volume…..


Revolving around the x axis
Revolving around the discs to find the volume…..x axis


Region bounded between y 1 x 0
Region bounded between discs to find the volume…..y = 1, x = 0,

y = 1

x = 0


Volume generated between two curves
Volume generated between two curves discs to find the volume…..

y= 1


Area of cross section
Area of cross section.. discs to find the volume…..

f(x)

g(x)


dx discs to find the volume…..


Your turn region bounded between x 0 y x
Your turn: Region bounded between discs to find the volume…..x = 0, y = x, 


Region bounded between y 1 x 1
Region bounded between discs to find the volume…..y =1, x = 1


Region bounded between y 1 x 11
Region bounded between discs to find the volume…..y = 1, x = 1


Region bounded between
Region bounded between discs to find the volume…..


Around the x axis set it up
Around the discs to find the volume…..x axis- set it up


Revolving shapes around the y axis
Revolving shapes around the discs to find the volume…..y axis


Region bounded between1
Region bounded between discs to find the volume…..


Volume of one washer is
Volume of one washer is discs to find the volume…..


Calculate the volume of one washer
Calculate the volume of one washer discs to find the volume…..


And again region bounded between y sin x y 0
And again…region bounded between discs to find the volume…..y=sin(x), y = 0.


Region bounded between x 0 y 0 x 1
Region bounded between discs to find the volume…..x = 0, y = 0,  x = 1,


Worksheet 5 delta exercise 16 5
Worksheet 5 discs to find the volume…..Delta Exercise 16.5


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