Approximating a Solid with Known Cross Sections

For a solid with known cross section, we can multiply the depth and the area of each cross section, and adds them together to find the total volume of the solid.

We investigate a solid whose base is bounded by the functions of y = sin(x) and y = cos(x), 0.25 π ≤ x ≤ 1.25 π, the cross sections perpendicular to the x-axis are equilateral triangles. The reason of choose this solid is that is a relatively complex object but is generated by two simple equations that we are familiar with. We can suppose this solid is a model of an island.

It is feasible to use the volume by cross section method to calculate total volume of the solid.

We can calculate the side length of each equilateral triangle shaped slice, and then calculate the area of each equilateral triangle. Further, we compute the volume of each slice by multiplying the area and the depth of the slice. By summing up volumes of all the slices, we get the total volume of the solid. The detailed calculation process is shown in the following table:

From the table above, we got the approximation of the volume is 1.358 cubic inches. We can use the following integral to calculate the actual volume of the solid. \[V=\frac{\sqrt{3}}{4} \times ∫_{\pi/4}^{5\pi/4}{(sin⁡(x)-cos⁡(x))^2}dx\] \[=\frac{\sqrt{3}}{4}\pi\] \[=1.360\] The actual volume of the solid is 1.360 cubic inches. The difference between the approximated volume and the accurate volume is 1.360-1.358=0.002 cubic inches. The relative error is 0.002/1.360 ≈ 0.15%.

Comments

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...

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...

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...

MA 391 Composition and Communication

Search This Blog