Applications of integration
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Applications of Integration. Volumes of Revolution Many thanks to http:// / 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

Method of discs

Take this ordinary line

Revolve this line around the x axis



We form a cylinder of volume

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



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…



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


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..




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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