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

Applications of Integration

Volumes of Revolution

Many thanks to

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


Method of discs

Method of discs


Take this ordinary line

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

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

2

5


If we stack all these slices

If we stack all these slices…

We can sum all the volumes to get the total volume


Applications of integration

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

The volume of one section:

Volume of one slice =


Applications of integration

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:

r = y value

h = dx

Volume of one slice =


Volume of cucumber

Volume of cucumber…

Area of 1 slice

Thickness of slice


Take this function

Take this function…

and revolve it around the x axis


Applications of integration

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…


Revolve it

Revolve it…


Christmas bell

Christmas bell…


Divide the region into strips

Divide the region into strips


Form a cylindrical slice

Form a cylindrical slice


Repeat the procedure for each strip

Repeat the procedure for each strip


To generate this solid

To generate this solid


A polynomial

A polynomial


Regions that can be revolved using disc method

Regions that can be revolved using disc method


Regions that cannot

Regions that cannot….


Model this muffin

Model this muffin.


Washer method

Washer Method


A different cake

A different cake


Slicing

Slicing….


Making a washer

Making a washer


Revolving around the x axis

Revolving around the x axis


Region bounded between y 1 x 0

Region bounded between y = 1, x = 0,

y = 1

x = 0


Volume generated between two curves

Volume generated between two curves

y= 1


Area of cross section

Area of cross section..

f(x)

g(x)


Applications of integration

dx


Your turn region bounded between x 0 y x

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


Region bounded between y 1 x 1

Region bounded between y =1, x = 1


Region bounded between y 1 x 11

Region bounded betweeny = 1, x = 1


Region bounded between

Region bounded between


Around the x axis set it up

Around the x axis- set it up


Revolving shapes around the y axis

Revolving shapes around the y axis


Region bounded between1

Region bounded between


Volume of one washer is

Volume of one washer is


Calculate the volume of one washer

Calculate the volume of one washer


And again region bounded between y sin x y 0

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


Region bounded between x 0 y 0 x 1

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


Worksheet 5 delta exercise 16 5

Worksheet 5Delta Exercise 16.5


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