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