Skip to main content

    In mathematics, the calculation of volume is an indispensable part. However, in real-life construction and various productions, the volume we require to calculate cannot be simply calculated like: long * width * high. So for all kinds of odd-shaped parts, if we want to know its volume (for example, how much material is needed to make a certain item), one of the ways is to divide it into many small cross-sections and perform step-by-step calculations ( You can also put it directly into a container filled with water, and then look at the volume of water discharged, I think this method is simpler, but not everything can be put in the water, is’t it?).

    To give a very simple example, to calculate the volume of a dumpling, direct calculation may not be very simple, but we can divide it into many parts longitudinally, so that each part looks like a triangle. By calculating the volume of each part, Then add up to get a relatively accurate value. The more data points, the closer to the real volume.
    I used two simpler equations: y=0.5\sin x and y=-0.5\sin x, They can intersect a lot of "dumplings"
    The intersection of the two functions is at (kπ, 0), k ∈ 整数集. So We take out one of the "dumplings":
    Then its overall length is from (0, 0) to (π, 0), we divide it into 15 parts to calculate, assuming that the cross section of each part is a regular triangle. The height of each serving is 0.20943.


    Modeling through these data, we can get this

    We add up all the volumes and we can get that the volume of "dumplings" is about 1.36026.
    
    So, next calculate the real volume of this "dumpling", we use the function 
∫ (0.5sinx-(-0.5sinx))*(sinπ/3)*(0.5sinx-(-0.5sinx))dx from 0 to π to calculate, The function in the middle can be simplified to sinx*sinx*(sinπ/3). After calculation, the final score is 1.36034. The two scores are very close. This is a very simple and easy to show example. Relatively speaking, it is also a special but very common Chinese food, so I hope it will help to understand this method.
    
Labels:

Comments

Post a Comment

Popular posts from this blog

Do Over: Integration Over a Region in a Plane

Throughout the semester we have covered a variety of topics and how their mathematical orientation applies to real world scenarios. One topic we discussed, and I would like to revisit, is integration over a region in a plane which involves calculating a double integral. Integrating functions of two variables allows us to calculate the volume under the function in a 3D space. You can see a more in depth description and my previous example in my blog post, https://ukyma391.blogspot.com/2021/09/integration-for-over-regions-in-plane_27.html . I want to revisit this topic because in my previous attempt my volume calculations were incorrect, and my print lacked structural stability. I believed this print and calculation was the topic I could most improve on and wanted to give it another chance. What needed Improvement? The function used previously was f(x) = cos(xy) bounded on [-3,3] x [-1,3]. After solving for the estimated and actual volume, it was difficult to represent in a print...
Post a Comment
Read more

Do Over: Ruled Surfaces

Why to choose this project to repeat For the do over project, I would like to choose the ruled surfaces. I don't think my last project was creative, and the 3D printed effect was not very satisfactory. In the previous attempts, all the lines are connected between a straight line and a circle. This connection structure is relatively uncomplicated. The printed model has too many lines, resulting in too dense line arrangement. The gaps between lines are too small, and the final effect is that all the lines are connected into a curved surface, which is far from the effect I expected. What to be improved In this do over project, I would like to improve in two aspects. Firstly, a different ruled surface is chosen. In the previous model, one curve is a unit circle on the \(x-y\) plane, and the ruled surface is a right circular conoid. In this do over project, it is replaced by two border lines. Each borderline is in the shape of an isosceles right triangl...
Post a Comment
Read more

Ruled Surfaces : Trefoil

Ruled Surfaces : Trefoil A ruled surface is a surface that consists straight lines, called rulings, which lie upon the surface. These surfaces are formed of a set of points that are "swept" by a straight line. This is relatively intuitive once you see a good visual, but can be a bit abstract without that concrete example. A very basic example of a ruled surface is a cylinder; if we have a straight line and move it in a circle we create a cylinder made entirely of straight line. Note that the surface will only be a cylinder if all the lines are parallel. If the lines are not parallel we can create hyperboloids and cones depending on how much we have rotated. The rotation we are describing here is not a simple turning action, but more of a twisting motion—less like rotating a can by turning it and more like wringing out a washcloth by twisting it. Specifically, a cylinder is essentially two circles connected by rulings, if we keep one of the circles...
Post a Comment
Read more