Skip to main content

Integration for over regions in the plane

With the continuous learning of the calculus course, it is no longer limited to a single plane. By performing quadratic integration, we have obtained a calculation method that can calculate the volume. From plane to three-dimensional, this is a new breakthrough. Simply put, it is to perform two integral operations, but it brings us new knowledge and challenges.

I really like the kind of "things" that burst out from an angle in front of you (similar to some skills in some games). So I chose two very simple equations to make them have a certain angle: y=2x and y=0.5x. Since the two equations will extend indefinitely, we also give him an interval. x=2 and y=2. Then the bottom of the entire graph has been determined, as shown in the figure:


Then we will divide this area a little bit more carefully, because it is a 2x2 square, then we will divide it into 100 small squares, so that the length of each side is 0.2. We use the point in the upper right corner as the benchmark. This point is within the angle between the two equations, and it belongs to a part of the volume (the point where the blue line passes is also counted, but the point where the green line passes first does not add to the volume. within.)

Then proceed to the next step. Now that we have determined the bottom, we need to find the height of each small square. I chose the equation:

Because I want to try what the root sign will be in this kind of graph in addition to various squares. And it will not cause the point (2, 2) to be too high and difficult to destroy the beauty. So let's first calculate the points obtained in this way, the height of each square and their volume.

Then the overall picture looks like this
Then after calculation, add all the rectangles together (a total of 55 rectangles, but fortunately the table can be directly calculated: D). So in the end, the volume I got was 2.659021.

So let's calculate its real volume, the way should be:
Then since the entire graph is composed of four sides, I divide the equation into two parts to calculate: x from 0 to 1 and x from 1 to 2. The equation expression is：


It can be seen that the number of 2.298 is much less than 2.659, which does not seem to be very close. I have also performed many calculations and checked whether the size and equation of each volume of the model are wrong, but it seems that this is the case. The main reason should be that the height of the entire volume increases as x and y increase. Think of it as a slope. This is different from the example we did before. The previous curve does not increase all the time, but has an ups and downs function. Then if I define it according to the upper right corner of the small square, every cylinder will It is a little bigger than it should be, instead of being big and small, a very similar value can be obtained by complementing each other. So the volume obtained by this method will be larger than its real volume.

This figure should be very interesting, because its center should be a straight line (x=y) but it is not a plane. Unfortunately, due to the limitation of volume, it cannot be displayed. Although it looks like a plane, it does Not a plane.

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

Minimal Surfaces

Minimum surfaces can be described in many equivalent ways. Today, we are going to focus on minimum surfaces by defining it using curvature. A surface is a minimum surface if and only if the mean curvature at every point is zero. This means that every point on the surface is a saddle point with equal and opposite curvature allowing the smallest surface area possible to form. Curvature helps define a minimal surface by looking at the normal vector. For a surface in R 3 , there is a tangent plane at each point. At each point in the surface, there is a normal vector perpendicular to the tangent plane. Then, we can intersect any plane that contains the normal vector with the surface to get a curve. Therefore, the mean curvature of a surface is defined by the following equation. Where theta is an angle from a starting plane that contains the normal vector. For this week’s project, we will be demonstrating minimum surfaces with a frame and soap bubbles! How It Works Minimum surfac...
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