Finding Volumes of Revolution

By Gerardo Gonzalez

Procedure

• Step1:Locate a symmetric, but irregular shaped object.

I chose this one: Picture of wine glass needs to be uploaded.

Step2:Using a Caliper to find approximate measurements

• I first measured the height of the irregular object. The height was 158.75mm• I then got a piece of graph paper and drew a line that was 158.75mm in length. • I then divided 158.75mm into 25 measurements that were 6.35mm apart from

each other. • Interval:6.35• Total measurements:25 • My x values are the length of the wine laying down sideways on the x axis.• For the Y values I used a caliper, measured and divided the width of the wine

glass by 2 to get half the height of the whine glass. I divide it by two so I can use the disk method to revolve it around the x axis. This is one way we can figure out the approximate volume of the wine glass.

• There is one slight problem. The disk method uses a function to find the volume of revolution. I don’t have a function, but I do know that I can incorporate the disk method into the trapezoid rule. By doing this I will then be able to find the approximate volume of the wine glass.

• I applied the Trapezoid rule with the disk method incorporated to find the volume of the solid.

Side note: When you use the disk method with the Trapezoid rule you are able to revolve your y values around the x axis giving you a 3 dimensional area.• In the following slide I show the trapezoid rule.• I then show how to incorporate the disk method into

the trapezoid formula. • This then results in calculating the approximate

volume of the wine glass.

Step3:Applying the Trapezoid Rule with the Disk method incorporated

Finding the wine glass volume by using Trapezoid Rule with Disk method incorporated

Trapezoid   Rule : ¿2 2 2 2 2 2

0 2 3 23 24 25

1 1( ) ( )[ ( ) ( ) ( ) ... ( ) ( ) ( ) ]

2 2

b aT n f x f x f x f x f x f x

n

Actual Calculationsb=158.75mma=0n=25

158.75 0x= 6.35

25

b a mmmm

n

¿Side note: The ½ has already been distributed inside the brackets.

3Approximate volume of wine glass:490361.1756mm

Step 4: Use Mathematica to plot a 3D model of the wine glass.