# A guide to use online tools to perform mathematical modeling

### By Owen Cheong

(Math, Physics, Chemistry tutor at The Edge Learning Center)

Many students choose to do mathematical modeling in their Math IA. In this article, I will talk about how to use Desmos to model a curve by running a regression.

Desmos is a popular online graphing tool. You don’t need to install any applications on your computer in order to use it. You can access the tool by clicking the following url: www.desmos.com/calculator

In the following tutorial, I will show you how to use this online tool to estimate the volume of this beautiful cherry blossom vase by using regression and integration.

### Step 1: Insert the image by clicking “+” sign on the top left corner

We need to rotate the image such that the edge of the vase is a function that passes through the vertical line test. To do this, we just need to input “-pi/2” for the angle.

### Step 3: Translate the graph by adjusting the center

Modify the center to (4,4,-0.06) to make the bottom center of the vase is at the origin.

### Step 4: Plot several points on the edge of the vase that lies on the first quadrant

We can do this by inserting a table by clicking the “+” sign on the top left hand corner. This time, we should select the ‘table’. After that, we can input several coordinates that lie on the edge of the vase. You can input the following data set:

 X1 Y1 0.01 0.2 0.1 0.8 0.3 1.45 1 2.28 2 3.03 3 3.34 4 3.2 5 2.68 6 1.87 7 1.38 8 1.56 8.2 1.65

### Step 5: Run a polynomial regression to find the best-fit curve

We need to find a smooth curve that could connect all the dots. It can be achieved by running a polynomial regression. In this case, I would like to perform a polynomial regression with degree 7. Here is what you need to enter for our regression model:

After that, the coefficients of the polynomial functions will be calculated. So we can conclude that the equation of the best-fit curve is

### Step 6: Estimate the volume of the flask using integration

By applying the formula of the disk method for the volume of revolution, we can deduce the volume of the scaled model is

The true volume of the blossom vase depends on the scale ratio of the model. If 1 unit in the x-y plane represents centimeters in the actual model, then the volume ratio would be . Therefore, we can estimate the true volume of the vase is about  cubic centimeter.

These steps are completed in the following saved task:

https://www.desmos.com/calculator/n43au6zyol