**3D Printing: Applications and The Math Involved**

What is 3D Printing?

The process of making a 3D object from a digital file

Additive process

The Math Behind 3D Printing

Fubinis Theorem

Summation Applications

3D Printing & Economics

Estimations about 3D printing's impact

Growth is exponential

**How is 3D printing used in the real world?**

**What industries is it most popular in?**

Theorem

: if f (x,y) is continuous over the region R = [a,b] x [c,d] defined by a </= x </= b and c </= y </= d then:

How Does this Apply to Calculus?

Fubini’s Theorem states that an object of n dimensions can be represented as a spectrum of layers of shapes of (n-1)-dimensional layers.

Simply means that any 3 dimensional shape can be represented as layers of 2 dimensional shapes and the area can be found this way as well

Contour Graph Problems!

Average Value of a Function

Anti derivative

Y av=

for a continuous function ƒ on [a,b]

Goal: The add the areas of stacked circles (multiplied by a height) to find the total volume, using summation.

Since the change is constant at ∆3,

we know that the summation formula is cubic.

Solve to find:

a=1/6

b=1/2

c=1/3

Plug several different n values into original equation:

Arrive at Formula:

Starting with with 0 mm, we're looking to find the sum of values up to 100 mm for the function: f(x)=x^2

Sum=338,350

338,350*π=1,062,957.87

(Equivalent of half circle)

Multiply by 2, find total area of layers that make up sphere is 2,125,915.75 mm squared.

Height of cylinder is 1, so volume is 2,125,915.75 mm cubed (2125.92 cm cubed).

Second Approach: Rotating Functions about the x-axis

Basic Principle: Have to examine how things change

Area along X axis

Area along the Y axis

Total Area under 3D Graph

**Just how popular has it become in the industrial world as a whole?**

There are three industries that 3D printing is most popular in

Medicine

Automotive

Culinary

Organs and Prosthesis

The Urbee 2

The Urbee 2 is a fully functional car that is made entirely out of parts made from 3D printing.

Food

Chefs use edible materials to print interesting designs

3D printing is extremely popular...

Using math

, we can see that the rate of the increase of

3D printer

sales in recent years, is extemly close to the rate of the increase in sales of the

printing press

during the

industrial revolution

.

This proves that 3D printing could be

revolutionary

in the modern industrial world.

3D Printer Calculations

2007-2011

Analysis!

3D Printer Sales

Step 2: Find regression curve that fits best

Step 3: Take first derivative of the equation of the regression curve.

Step 4: Choose a point of comparison and plug in the year to the first derivative.

Record Results!

Printing Press Calculations

1820-1824

Printing Press Sales

Step 2: Find regression curve that fits best

Step 3: Take first derivative of the equation of the regression curve.

Step 4: Choose a point of comparison and plug in the year to the first derivative.

Record Results!

Solution

Now let's analyze the results...

Source: Additive Manufacturing and 3D Printing State of the Industry.

*Sales: Hundreds of companies that purchased a 3D printer.

The math involved in determing the popularity of 3D printing.

Let's find the rate of change during the 4th year (2010) by finding the first derivative at this point.

Step 1: Graph the data for the number of sales recorded in each year of data collection.

2007 2008 2009 2010 2011

1820 1821 1822 1823 1824

*Sales: Number of Companies (hundreds)

Step 1: Graph the data for the number of sales recorded in each year of data collection.

Source: Historyguide.org

Solution

Let's find the rate of change during the 4th year (1823) by finding the first derivative at this point.

Since this is the first derivative, the units for this number are hundreds of companies per year per year, we can conclude that:

The number of companies that owned printing presses was growing at a rate of

26,1034 companies per year per year in 1823

.

The number of companies that own 3D printers was growing at a rate

1,044,450 companies per year per year in 2011.

From this number we can see that:

Now what do these two numbers tell us?

2,610.34 companies per year per year in 1823.

The rates of change that we calculated come out to be:

1,044,450 companies per year per year in 2011.

The industrial world has grown in the past two years, so much so that the

World Federation of Exchange

has estimated that there are about

400 times

more companies in the United States in 21st century than in 19th century.

This means that if we want to compare these two numbers on the same scale we need to divide the rate of change in 2011 by 400.

and

1,044,450 / 400 = 2,611.125

When compared on the same scale, the two rates of change differ only by 0.785 companies!

We found the number of companies in the United States that owned these two 'printers' over the span of the five years when they were most popular, and when their sales saw the most growth from the 4th to 5th year.