Introduction
By finding the volume of a solid that has known cross-sections we go way back to some of our very early years of math. With cross-sections, we visit some very common shapes such as: triangles, semi-circles, rectangles and squares, and other polygons. Today we will be specifically looking at triangles. The process for finding the volume of a solid is rather simple. We are going to sketch the cross-section of the function and then express the area as a function. After, we are going to determine the limits of integration and integrate the definite integral.
Cross Section?
There are a variety of different shapes that you can conduct the volume of a solid using cross-sections. For simplicity, I am going to conduct the volume of a solid using triangles as my cross-sections. In my example, I am going to be using two functions. The first function is \(g(x)=\frac{1}{2}x^2\) and the second function is \(f(x)=x^2\).
First, let's analyze this graph of \(g(x\)) and \(f(x\)).
Although the functions do not intersect, we are still able
to take the cross-sections to find the volume of the
solid. We will be dividing our solid into 10 steps which
would result in 10 equilateral triangles. By stacking
equilateral triangles on top of one another and adding the
volumes of each triangle up, we can approximate the volume
of the solid. From the graph, we are going to be taking
the cross-sections between the interval of \(1≤x≤2\). This
would result in our triangles to have a 0.1 width for
every step. Since we are using equilateral triangles, we
will have to use the area of a equilateral triangle when
we approximate.
Approximating and Computing with Cross Sections
As stated previously, to find the volume of a solid that has known cross-sections as triangles is rather simple. We are going to stack 10 equilateral triangles on top of each other, and then add the volume of each triangle up to approximate the total volume of the solid. Since our solid is bounded between \(1≤x≤2\) we are going to have 10 equilateral triangles with width 0.1. By taking the area of an equilateral triangle \[Area = \frac{\sqrt3}{4}s^2\] where \(s\) is the length of the sides. Then, once we compute the area of the equilateral triangle we will then multiply it by 0.1 since that is the length of each step, and thus the width of each triangle. By doing this, we will get the total volume of the solid. Below is the table for our approximation.
This gave us an approximation of \(Volume = 0.593
units^3\)
Next, to calculate the accurate volume of the solid,
we need to integrate. Using the formula from earlier,
\[Area = \frac{\sqrt3}{4}s^2\] we will slightly modify
it to include the two functions. The full integral is:
\[\frac{\sqrt3}{4}\int_a^b(f(x)-g(x))^2dx\] In this
equation \(g(x\)) is
\(\frac{1}{2}x^2\)
and \(f(x\)) is
\(x^2\)
which are integrated from the range 1 to 2.
Substituting the functions and the bounds in,
we get the integral:
\[\frac{\sqrt3}{4}\int_1^2((\frac{1}{2}x^2)-(x^2))^2dx\]
Evaluating the integral we then get:
\[\frac{\sqrt3}{4}\int_1^2((\frac{1}{2}x^2)-(x^2))^2dx
\rightarrow
\frac{\sqrt3}{4}\int_1^2(-\frac{x^2}{2})^2dx
\rightarrow
\frac{\sqrt3}{4}\int_1^2(\frac{x^4}{4})dx
\rightarrow \frac{\sqrt3}{4}(\frac{x^5}{20})\]
\[\frac{\sqrt3}{4}(\frac{x^5}{20}) \rightarrow
\frac{\sqrt3}{4}((\frac{2^5}{20})-(\frac{1^5}{20})
\rightarrow \frac{\sqrt3}{4}\cdot\frac{31}{20}
= \frac{31\sqrt3}{80}\] \[= 0.67116 units^3\]
Why These Functions?
These functions were chosen for a few reasons. For someone who is just starting to learn this topic about finding the volume of a solid with known cross-sections, understanding the concept is more important than remembering the integration, etc. Thus, I chose these functions due to the integration being rather easy. This would allow someone to focus more on the concept, and being able to further understand cross-sections than focusing on the integration. Also, I chose to use equilateral triangles as my cross-sections due to my opinion of triangles being an easy shape to understand the concept with. The area of an equilateral triangle is rather easy too. I believe that using equilateral triangles allow for someone to better grasp the concept than using a more difficult shape for my cross-sections. I believe that the hardest part when learning cross-sections as well as most of calculus is being able to visualize the concept. Being a visual learner myself, being able to visualize a concept helps a lot, thus using triangles to keep simplicity was imperative when choosing my cross-functions and the functions used allow for someone to easily visualize the solid.
By finding the volume of a solid that has known cross-sections we go way back to some of our very early years of math. With cross-sections, we visit some very common shapes such as: triangles, semi-circles, rectangles and squares, and other polygons. Today we will be specifically looking at triangles. The process for finding the volume of a solid is rather simple. We are going to sketch the cross-section of the function and then express the area as a function. After, we are going to determine the limits of integration and integrate the definite integral.
Cross Section?
There are a variety of different shapes that you can conduct the volume of a solid using cross-sections. For simplicity, I am going to conduct the volume of a solid using triangles as my cross-sections. In my example, I am going to be using two functions. The first function is \(g(x)=\frac{1}{2}x^2\) and the second function is \(f(x)=x^2\).
As stated previously, to find the volume of a solid that has known cross-sections as triangles is rather simple. We are going to stack 10 equilateral triangles on top of each other, and then add the volume of each triangle up to approximate the total volume of the solid. Since our solid is bounded between \(1≤x≤2\) we are going to have 10 equilateral triangles with width 0.1. By taking the area of an equilateral triangle \[Area = \frac{\sqrt3}{4}s^2\] where \(s\) is the length of the sides. Then, once we compute the area of the equilateral triangle we will then multiply it by 0.1 since that is the length of each step, and thus the width of each triangle. By doing this, we will get the total volume of the solid. Below is the table for our approximation.
These functions were chosen for a few reasons. For someone who is just starting to learn this topic about finding the volume of a solid with known cross-sections, understanding the concept is more important than remembering the integration, etc. Thus, I chose these functions due to the integration being rather easy. This would allow someone to focus more on the concept, and being able to further understand cross-sections than focusing on the integration. Also, I chose to use equilateral triangles as my cross-sections due to my opinion of triangles being an easy shape to understand the concept with. The area of an equilateral triangle is rather easy too. I believe that using equilateral triangles allow for someone to better grasp the concept than using a more difficult shape for my cross-sections. I believe that the hardest part when learning cross-sections as well as most of calculus is being able to visualize the concept. Being a visual learner myself, being able to visualize a concept helps a lot, thus using triangles to keep simplicity was imperative when choosing my cross-functions and the functions used allow for someone to easily visualize the solid.
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