Applications of Integration

Volumes of Revolution

Many thanks to

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

Method of discs

Take this ordinary line

2

5

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

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…

Each slice would have a thickness dx and height y.

25-5

The volume of one section:

r = y value

h = dxVolume 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…..

Radius = y

Area =

Thickness of slice = dx

Volume of one slice=

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