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AP CALCULUSMrs. DiCosmo
Volumes of Solids of Revolution:
• Disk Method• Washer Method
You will be able to calculate volumes of irregular shaped solids
Some of the Professional fields that are using this particular concepts of Integral Calculus
Containers and Packaging
Construction
MRI & CAT scan
Idustrial Designs
Reminder !!!!!!!!
n
k
b
akn
dxxfxxf1
)()(lim
n
abx
xkaxk
Definition of a Definite Integral
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
• Consider the area between two functions rotated about the axis
• Now we have a hollow solid
• We will sum the volumes of washers
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.
ANY QUESTIONS ?
HOMEFUN !!!
Pg. 423 / ex. 3-13 all
http://www.learnerstv.com/Free-maths-Video-lectures-ltv295-Page1.htm
Helpful Links:
https://www.khanacademy.org/math/calculus/integral-calculus
Sources:http://www2.bc.cc.ca.us/resperic/Math6A/Lectures/ch6/2/washer.htm
http://tutorial.math.lamar.edu/Classes/CalcI/VolumeWithRings.aspx
http://math.hws.edu/~mitchell/Math131S13/tufte-latex/Volume2.pdf
https://www.google.com
http://www.learnerstv.com/Free-maths-Video-lectures-ltv295-Page1.htm
https://www.khanacademy.org/math/calculus/integral-calculus
Assessment
