Skip to main content

Approximating a Solid by Rectangular Prisms

About integration for a function defined on a region

If a solid can be modeled by an integrable function of two variables over a general region in the plane, we can use integration to find the volume of the solid. Let’s consider a region \(D\) on the \(x-y\) plane, which is bounded by two vertical lines and two functions of \(x\). The two vertical lines are \(x=a\) and \(x=b\) respectively, and the two functions of \(x\) is \(y=u(x)\) and \(y=v(x)\). The integrable function is \(f(x,y)\). The integration to find the volume of the solid can be calculated as following: \[V=\int^b_a{\int^{v(x)}_{u(x)}{f(x,y)dy}dx}\] The double integral \(\int^{b}_{a}{\int^{v(x)}_{u(x)}{f(x,y)dy}dx}\) of positive \(f(x,y)\) can be regarded as the volume under the surface \(z=f(x,y)\) over the region \(D\). We can use the Riemann sum to approximate the integral: \[\sum_{i,j}{f(x_i,y_j)}\Delta x \Delta y \] Each term in the Riemann sum is the volume of a thin box with base \(\Delta x \times \Delta y \) and height \( f(x_i,y_j)\).

The solid to be analyzed

The function we chose is a hyperbolic paraboloid. The surface defined by the hyperbolic paraboloid is also called saddle, as it is visually similar to a saddle for horse-riding. The hyperbolic paraboloid function is defined by \[z=f(x,y)=1+2x^2-2y^2\]
The region on the \(x-y\) plane is bounded by two vertical lines and two functions of \(x\). The two vertical lines are \(x=-0.6\) and \(x=0.6\) respectively, and the two functions \(y=u(x)\) and \(y=v(x)\) are hyperbolas: \[ y=u(x)=-\sqrt{0.28+x^2}, -0.6 \le x \le 0.6\] \[ y=v(x)=\sqrt{0.28+x^2}, -0.6 \le x \le 0.6\]
The solid can be defined by the saddle surface and the bounded region as the following picture shows:
This solid was chosen for the following reasons:
1. The function of the saddle surface has a concise form, so that we can focus on the concept of approximating the volume by double integral. The saddle surface we chose is not only important from the mathematical point of view, but also popular such as a saddle for horse riding in our daily lives. The Gaussian curvature, which is the product of the two principal curvatures \(Κ = κ_1κ_2\), is negative for a saddle shaped surface.
2. The region on the \(x-y \) plane is not a simple rectangle, but a somewhat irregular region bounded by two vertical lines and hyperbolas. By using numerical approximation, i.e. the Riemann sum, we can easily calculate the volume of the solid over the region.
The functions we chose made a shape with irregular surface and base region. This helps us to understand the concept and the calculation process of using the double integral to find the volume of the solid.

Compute the volume of approximation

We can compute the volume of solid approximated by rectangular prisms. The calculation process is shown in the following table
From the table, the volume of approximation \(V'\) is 1.488 cubic inches.
The 3D shape of the approximation is shown in the following picture:
If more rectangle prisms are used, it is possible to get more accurate approximation.

Compute the actual volume

The actual volume can be calculated by double integration: \[V=\int^{-0.6}_{0.6}{\int^{\sqrt{0.28+x^2}}_{-\sqrt{0.28+x^2}}{(1+2x^2-2y^2)dy}dx}=1.493\] The actual volume of the solid is 1.493 cubic inches. The difference between the approximated volume and the accurate volume is \( \Delta V= |V-V'|=1.493-1.488=0.005\) cubic inches. The relative error is \( \frac{\Delta V}{V}= 0.005/1.493 ≈ 0.33\% \).
From the above example, we can better understand the concept and the numerical approximation of the double integral of a function over a region on 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

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