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