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# Applications of Integration - PowerPoint PPT Presentation

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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Volumes of Revolution

Many thanks to

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

Revolve this line around the x axis

2

5

We form a cylinder of volume

If we stack all these slices… sections

We can sum all the volumes to get the total volume

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

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 section: revolve this curve around the

r = y value

h = dx

Volume of one slice =

Volume of cucumber… revolve this curve around the

Area of 1 slice

Thickness of slice

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=

Area =

Thickness of slice = dx

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

Revolve it… discs to find the volume…..

Christmas bell… discs to find the volume…..

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

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

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

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

A polynomial discs to find the volume…..

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

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

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

Washer Method discs to find the volume…..

A different cake discs to find the volume…..

Slicing…. discs to find the volume…..

Making a washer discs to find the volume…..

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

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

y = 1

x = 0

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

y= 1

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

f(x)

g(x)

dx discs to find the volume…..

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

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

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

Region bounded between discs to find the volume…..

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

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

Region bounded between discs to find the volume…..

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

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

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

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

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