7.3 volumes by cylindrical shells
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Transcript of 7.3 volumes by cylindrical shells
Volumes of Solids of Revolution: • Disk Method• Washer Method
Make Sure You Remember Process for Calculating Area
Divide the region into n pieces.
Approximate the area of each piece with a rectangle.
Add together the areas of the rectangles.
Take the limit as n goes to infinity.
The result gives a definite integral.
General Idea - Slicing
1. Divide the solid into n pieces (slices).
2. Approximate the volume of each slice.
3. Add together the volumes of the slices.
4. Take the limit as n goes to infinity.
5. The result gives a definite integral.
Disk Method
Volume of a SliceVolume of a cylinder?
h
r
2V r h
What if the ends are not circles?
A
V Ah
What if the ends are not perpendicular to the side?
No difference! (note: h is the distance between the ends)
Volume of a Solid
1
lim ( )n
kn
k
V A x x
a xk b
A(xk)
( )slice kV A x x
x
( )b
aA x dx
The hard part?
Finding A(x).
Volumes by Slicing: ExampleFind the volume of the solid of revolution formed by rotating the region bounded by the x-axis and the graph of from x=0 to x=1, about the x-axis.
Here is a Problem for You:Find the volume of the solid of revolution formed by rotating the region bounded by the x-axis and the graph of y = x4, from x=1 to x=2, about the x-axis.
Ready?A(x) = p(x4)2= px8.
Washer Method
Setting up the Equation
Outer Function
InnerFunction
R
r
Solids of RevolutionA solid obtained by revolving a region around a line.
When the axis of rotation is NOT a border of the region.
Creates a “pipe” and the slice will be a washer.
Find the volume of the solid and subtract the volume of the hole.
f(x) g(x)
xk ba
NOTE: Cross-section is perpendicular to the axis of rotation.
2 2( ) ( )
b b
a aV f x dx g x dx
2 2( ) ( )
b
aV f x g x dx
Example:Find the volume of the solid formed by revolving the region bounded by y = (x) and y = x² over the interval [0, 1] about the x – axis.
2 2([ ( )] [ ( )] )b
a
V f x g x dx
1
0
222dxxxV
Here is a Problem for You:Find the volume of the solid of revolution formed by rotating the finite region bounded by the graphs of about the x-axis.
Ready?
So……how do you calculate volumes of revolution?
• Graph your functions to create the region.
• Spin the region about the appropriate axis.
• Set up your integral.
• Integrate the function.
• Evaluate the integral.