Finding the Volume of a Vase Using the Simpson’s Rule

The goal of this experiment was to find the volume of a symmetrical, but irregular shaped object (assuming it is solid) by using either the Simpson's or Trapezoidal Rule. The object chosen for this experiment was a glass vase. This can be seen in Figure 1. After the object was chosen, the height of the object was measured. The total height of the vase was 26.67cm. The total height was divided by 24, allowing the ∆x to be 1.11125cm. Starting at 0.00cm, a mark was made on the vase in 1.11125cm intervals. At each of these marks, the diameter was measured using calipers. Each measured diameter was then divided into two and the radius was found. The thickness of the glass was also measured using the calipers and this was subtracted from the radius. Figure 1 shows how the vase was measured using a measuring tape and calipers. Due to the fact that the calipers and the measuring tape had units of inches, all of the measurements were converted, using the conversion factor, to centimeters. The measured radius at each specific height can be seen in Table 1. Using these specific measurements, a graph was plotted in Maple. The x-axis of the graph represents the height of the vase, and the y-axis represents the measured radius. This graph can be seen in Figure 2. The calculations that were completed to find the volume can be seen below. After all the calculations were complete, it was found that the volume of the vase is 1866.237cm³.


Calculations:

  • From the disk method, it is found that:


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  • Using this information, the Simpson’s rule was manipulated to calculate volume.

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  • Using the f(x) values that were calculated in the first part of this experiment, the volume was found. The equation can be seen below:

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Figure 1. The images seen below represent the way the vase was measured using calipers and a measuring tape.

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Table 1. This table contains all of the measurements of the vase.

Height (cm)

(x-axis)

Radius (cm)

(y-axis)

0

3.429

1.11125

5.26034

2.2225

5.8801

3.33375

6.05

4.445

6.1214

5.55625

6.1214

6.6675

6.1

7.77875

5.9944

8.89

5.76

10.00125

5.5

11.1125

5.14571

12.22375

4.74853

13.335

4.32435

14.44625

3.88112

15.5575

3.59029

16.66875

3.3274

17.78

3.17135

18.89125

3.05435

20.0025

3.05435

21.11375

3.175

22.225

3.3909

23.33625

3.65

24.4475

3.99669

25.55875

4.3942

26.67

4.75615

Figure 2. The following is a graph of the height vs. the radius of the vase.

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