Applications of Integration Volumes of Revolution Many thanks to od/gallery/gallery.html.

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Presentation transcript:

Applications of Integration Volumes of Revolution Many thanks to od/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

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….. 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 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 between y=sin(x), y = 0.

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

Worksheet 5 Delta Exercise 16.5