Volumes of Solids of Revolution: • Disk Method • Washer Method
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Volumes of Solids of Revolution: • Disk Method • Washer Method
Make Sure You Remember Process for Calcula4ng 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 Slice Volume of a cylinder?
h
r
2V r hπ=
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
1lim ( )
n
kn kV A x x
→∞=
= Δ∑
a xk b
A(xk) ( )slice kV A x x= Δ
xΔ
( )b
aA x dx= ∫
The hard part?
Finding A(x).
Volumes by Slicing: Example Find 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.
y = x
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
SeAng up the EquaDon
Outer Function
Inner Function
R
r
Solids of RevoluDon A 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 b a
NOTE: Cross-section is perpendicular to the axis of rotation.
[ ] [ ]2 2( ) ( )b b
a aV f x dx g x dxπ π= −∫ ∫
[ ] [ ]2 2( ) ( )b
aV f x g x dxπ= −∫
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 dxπ= −∫
( ) ( )∫ −=1
0
222dxxxV π
V = π (x − x4 )dx0
1∫
V = πx2
2−x5
5"
#$
%
&'0
1
=310
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 revolu4on?
• Graph your functions to create the region.
• Spin the region about the appropriate axis.
• Set up your integral.
• Integrate the function.
• Evaluate the integral.
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