Skip to main content

Use The Washer Method to find out the volume.

It may not be difficult to figure out the volume of a cylinder, but when it is a relatively irregular or variable width volume, the calculation is much more troublesome. The Washer Method is one of the solutions to this situation.

When we mastered how to find the area of a quadratic function by dividing it into segments, people improved this method and applied it to finding the volume of a three-dimensional. Use the Washer/Shell method to find the volume. What I want to show today is the "Washer."



Think of an area, which is obtained by the intersection of two functions, and then this area enters a three-dimensional space and rotates around the y axis to obtain a volume.




The red line in the figure is 



The green line is 


The entire graphic rotates around the y axis to get a "cup", which will make it difficult for us to calculate the volume of this "cup", so we divide it into a circle and a circle of cushions, and calculate each one The volume of the mat can be added together to get an approximate volume.




This is the data obtained after dividing it into 10 parts. What is interesting is that the volume seems to be symmetrical (the volume of the first few pieces is the same after the latter). I have calculated many times, so I am sure that it is not a wrong number, it should be some kind of coincidence.\par

Therefore, we can get a similar volume through The Washer Method, as shown in the figure\par





So, through this method we can calculate that the volume is approximately 0.56676



We can get the volume of the intersecting area by calculating the volume of the two parts and then subtracting them.

Since it is rotating around the y axis, our equation is





After calculation, mine score is 0.57256

This result is very close, and we only need to divide it more carefully to get a more precise answer.


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