MA 391 Composition and Communication

Come see them on the seventh floor of Patterson Office Tower.

Think about you are cutting a steak to slide. What is so fantastic is that each slice of steak represents a cross-sectional area of the whole steak, which means when you put them together, you will get the volume of the solid steak. Cross Section is a topic that we learn in Calculus II; it is common in daily life. However, it is not easy to do it in Calculus for a beginner. So, in this following Blog, I will use a 3-D model and other visual ways to introduce how to calculate the volume of a 3-Dimensional solid by taking an area and building up from that area using known cross-sections. How to do this? I will use a example from calculus book to explain this. The flat base of a solid sits in the xy-plane in the region bounded by the graphs of y=0 and \(y=e^{−x}\) between \(x=0\) and \(x=1\).Find volume of this solid if cross-sections are perpendicular to the x-axis and they are squares. The following graph shows that the b...

To find the volume of a pyramid with square base we first have to find the dimensions of the pyramid. For this example, the sides of the base are 6 and the height of the pyramid is 4. The lengths of the lines going from the corner of the base square to the top of the pyramid is 5 (found from the distance formula between points (0,4) at the top of the pyramid and (3,0) the corner edge of the rectangle base. This isn't necesarry for the integral but it is helpful in constructing the pyramid in onshape. Next we need to find the equation of the area of a representative rectangle (in this case a square) inside the pyramid. To do this we use the area of a square formula where the sides are "a" length and the area is equal to \(a^2\). We want this in terms of "x" instead of "a", so we say that "a" is equal to "2x". This is because the length from the center of the representative rectangle to the edge of the pyramid is "x...

Solids of Known Cross Section : Toaster Calculating the volume of "irregular" solids&#8212as opposed to regular solids such as rectangular prisms, a square pyramid, etc.&#8212can be difficult. We have previously reviewed solids of revolution and a method of obtaining the volume of one such solid, specifically the method of volumes by cylindrical shells. This method is useful for finding the volumes of "round" or "circular" solids&#8212i.e. those that have been revolved about an axis; while the formulas for the volumes of spheres and cones are known we can easily approximate their volumes via the washer or shell methods, furthermore, the volumes of solids whose volumes are unknown&#8212like an egg, doorknob, or lampshade&#8212can also be easily approximated using these methods. We are now examining other solids that have not been revolved about an axis or are not "round" such as a roof or an eye glasses...

Solids with Known Cross Section      Cross sections help us visualize how multiple 2D planes can make up a 3D object. For example, a camping tent has cross sections with the shape of a triangle or rectangle (depending on if you are looking at the parallel or perpendicular angle) or a soccer ball has cross sections in the shape of a circle. The Egyptian Pyramids were even built one cross section square at a time. By breaking down a 3D object into simple 2D shapes, we can easily visualize a 3D model and calculate its volume.      For me personally, one of my favorite foods is ice cream and it’s to die for out of a waffle cone! But every time I go to an ice cream parlor, I struggle with the million-dollar question, waffle cone or cup? Off the top of my head, I choose a scoop in a cup because I think I’ll get more ice cream (which is the main goal of course!) but I just never know. To help figure this out, the function I chose to focus on is an absolute value f...

Finding the volume enclosed by a curve using a known cross section provides an alternative to using the solid of revolution method. In this scenario, the value of the curve at a given point could act as the side length of a polygon or the radius of a hemisphere. Likewise, many other options exist for this method, and any cross section allows for the computation of volume as long as the cross section has a feasible formula.  Every value of the function then goes into the area function of the cross section. Moreover, each cross section acts as an individual where the change in the independent variable acts as the width. Rather than finding the area under the curve, we find the volume of the sum of these cross section slices. When we have cross sections perpendicular to the x-axis we have a function of the x-variable, but when we have cross sections perpendicular to the y-axis, we have a function of the y-variable. For each slice, \(dx\) and \(dy\) act as the width of e...

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 ...

Circles aren't the only shape that can take a stab at guessing the volume of a solid. Any shape with the an area that can be calculated can be used to approximate the area of a solid. The method of approximation by cross section takes advantage of a certain shape to generate a model of 3-D structure. This concept is a more general form of the previously explored disks and washers. This time around, one function will be used as bound for a region that will be rotated around an axis. That function is: \[ f(x) = tanx \] bounded at the line y = 1. Approximation Using Squares For this example, squares will be used as the cross sections to create the approximation. The reason that this function was chosen was because it seemed like an interesting shape to rotate around the x-axis. It also seemed ammusing to utilize squares to approximate a trig function. It almost feels like trying to push a cube through a circle shaped hole, but maybe its approximation wi...

The equation I chose was \(\log_{10}x\) whose cross section is squares with side length x. My model is eleven horizontal square layers from y=0 to y=1.1. Each layer's side length is its smaller x value on the function.   The reason I chose \(\log_{10}x\) is to demonstrate the square cube law. The bottom layer's side length is \(\frac{1}{10}\) the side length of the top layer, but the top layer has 100 times the area of the bottom layer. This is because of the square-cube law. If a shape's nth dimension grows by a factor, k, to make a similar object, its mth dimension grows by a factor of \(\sqrt[n]{k^{m}}\). My model's first layer increased by a factor of 10 in the 1st dimension. Area is used in the second dimension, so the increase in area is \(\sqrt[1]{10^{2}}\) or 100. The model's layers have constant height, so its volume increases by the same factor as its area. If its height was proportional to its side length, though, the volume would increase by ...

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 cro...

This week, we are going over solids of known Cross-Sections! This is another method of integration, going along with the theme of last week. The solids of known Cross-Sections is very similar to the process of integration by Shells and Washers, since these are merely cases of solids of known Cross-Sections. In order to find the volume of a known Cross-Sections, we merely take the area of that shape on the 2-D plane, and then extend it to the 3-D plane by multiplying the area by the height, where the height represents how much you extend into the 3-D plane. Below is a simulation of what the cross-section actually looks like, whose creator is Michael Andrejkovics (a link to their Geogebra will be below!). Now that we have the foundation down, let us build! The function I chose to showcase solids of known Cross-Sections is not necessarily difficult, but a twist on a plain example of using. The function is: $f(x) = \sin(x)*\pi*e$. I originally wanted to use a more complex functi...

Introduction Volume can be found of anything solid. Knowing how to take the volume of a simple shape can help measure the volume of any shape or solid. How can this be done? Taking a solid and dividing it up into multiple parts, which are the cross-sections, and measuring the volume for each of the cross-sections separately. The cross-sections would be a shape we do know how to take the volume of easily like a square, rectangle, or oval. There is no limit on how many cross-sections can be used. Using more cross-sections can give an accurate and closer volume to what the solid actually is. To take a closer look at this lets look at the equations \[y = 1-x^2\] and \[y = -1+x^2\] When you graph these two functions it seems like a football from \(-1≤x≤1\) as below. To approximate this solid we will be using 10 rectangle cross-sections perpendicular to the x-axis. The graph above is in 2-dimensional to make this into 3-dimensional imagine cutting this footbal...

    I n mathematics, the calculation of volume is an indispensable part. However, in real-life construction and various productions, the volume we require to calculate cannot be simply calculated like: long * width * high. So for all kinds of odd-shaped parts, if we want to know its volume (for example, how much material is needed to make a certain item), one of the ways is to divide it into many small cross-sections and perform step-by-step calculations ( You can also put it directly into a container filled with water, and then look at the volume of water discharged, I think this method is simpler, but not everything can be put in the water, is’t it?).      To give a very simple example, to calculate the volume of a dumpling, direct calculation may not be very simple, but we can divide it into many parts longitudinally, so that each part looks like a triangle. By calculating the volume of each part, Then add up to get a relatively accurate value. The more ...