MA 391 Composition and Communication

Come visit them on the seventh floor of Patterson Office Tower! They will be replaced by the solids with known cross section soon.

Use The Washer Method to find out the volume. It may not be difficult to figure out the volume of a cylinder, but when it is a relatively irregular or variable width volume, the calculation is much more troublesome. The Washer Method is one of the solutions to this situation. When we mastered how to find the area of a quadratic function by dividing it into segments, people improved this method and applied it to finding the volume of a three-dimensional. Use the Washer/Shell method to find the volume. What I want to show today is the "Washer." Think of an area, which is obtained by the intersection of two functions, and then this area enters a three-dimensional space and rotates around the y axis to obtain a volume. The red line in the figure is  The green line is  The entire graphic rotates around the y axis to get a "cup", which will make it difficult for us to calculate the volume of this "cup", so we divide it into a circle and a circle of cushions, and...

The washer method is to slice an object into washer shaped slices, and then integrate over these slices to compute the volume of the object. By rotating an ellipse around the y-axis, we can generate a torus whose volume can be calculated using the washer method. The function of the selected ellipse is as following: \[\frac{(x-0.6)^2}{0.2^2}+\frac{(y-0.5)^2}{0.5^2}=1\] We can get a torus generated by rotating the selected ellipse around the y-axis. It is feasible to use washer shaped slices stacked along the y-axis to approximate the torus. We can calculate the area of the ring for each washer shaped slice, and then we compute the volume of the washer shaped slice by multiplying the area and the height of the washer shaped slice. The calculation process is shown in the following table: From the table above we...

Most math teachers I've had have been able to break down Calculus into two very broad categories: derivatives and integrals. What is truly amazing, is how much you can do with these two tools. By using integration, it is possible to approximate the shape of a 2-D function that is rotated around an axis. This solid created from the rotation is known as a solid of revolution. To explain this concept, we will take a look at the region bounded by the two functions: \[ f(x) = 2^{.25x} - 1 \] and \[ g(x) = e^{.25x} - 1 \] bounded at the line y = 1. This region is meant to represent a cross section of a small bowl. While it may not perfectly represent this practical object, the approximation will be quite textured, and will provide insight into how the process works. The region bounded by the two functions can be rotated around the y-axis to create a fully solid object. This is easy enough to talk about, but what exactly does this new solid look like? Is...

A solid of revolution is formed when an area in the plane is revolved around a line in the same plane. For example, Revolving a rectangle, produces a solid right circular cylinder. Revolving a semi-disk produces a solid spherical ball. The axis of revolution is the line around which the form is rotated. There are two methods to solve this question: Disks and Washers. This blog will introduce Washer Methods , which is well known as hollow soild revolution. In order to use washer methods, it could obtain two function and it could roated it either in \(x-axis\) or \(y-axis\). It is a demestraion of the solid roated around \(x-axis\). For example, we want to calculate the volume of the solid obtained by rotating the region bounded by the parabola \(y=x^2\) and the square root function \(y=\sqrt{x}\) around the \(x-axis\) In ...

Throughout one's mathematical journey, we come across this constant use of approximations. Approximations help us apply concepts in real-world activities since it becomes very difficult to show off some expressions (infinite decimal approximations for example). While you are in Calculus One, we are introduced to this concept of integrating a function. Integrating a function, let's say f(x)), is a method of finding the area underneath that curve on a graph. This application of area is very useful, especially when looking at graphs like: Distance vs. Time, Velocity vs. Time, and Acceleration vs. Time. The next natural thought after coming up with the area is “How can we extend this 2-D method into a 3-D world?”. Instead of looking at objects with a 2-D lens, once we get into Calculus 2 we can view things as 3-D. This leads up to going from finding the area of an object to finding the volume instead. How do we do this? We want to look at a function and rotate it along an axi...

Introduction We have learned the process for calculating the area of functions using integration, but what about calculating volume? Incorporating the third dimension (z-axis) seems rather challenging but calculating volume of three-dimensional solids using integration is a rather interesting concept. This concept uses cross-sections and certain methods such as the one we will be looking at today. Washer Method? The washer method can be used with two different scenarios. The first scenario just being a single function of \(y=f(x)\). The second scenario being the two functions \(y=f(x)\) and \(y=g(x)\). In my example, I am using two functions to show the washer method. My first function is \(y=x^3\) and the second function is \(y=\sqrt[3]{x}\). ...

The equations I chose for my example are \(e^{-x}\) and \(\ln x\). I used the shell method from 0 to 1 with ten divisions. I used the right hand rule for the approximation because if I used the left hand rule, the innermost shell would be infinitely long. Each shell is .1 inches thick and its height above the outermost shell is its radius plugged into \(e^{-x}\). Its depth below the outermost shell is its radius plugged into \(\ln x\).  The reason I chose these equations is to show how big of a difference an asymptote makes. The upper bounds of the graph cross the y-axis at y=1. The lower bounds never cross the y-axis. Looking at a graph of a function with an asymptote, you see the function get very close to the asymptote and is nearly touching it. The difference between the asymptote and the function gets smaller and smaller, but never reaches zero. My model shows how big of a difference crossing the y-axis is to almost crossing it. The height of each shell gets sligh...

Eggs and Shells An object of revolution is an object that is formed by taking the area under a curve(s) and rotating it about an axis. The volumes of such objects can be difficult to calculate; while others may be easier. Some objects whose volumes would be easy to calculate include a basketball, can of soup, or an icecream cone as these objects have known volume formulas—i.e. a sphere, cylinder, and cone, respectively. Other objects can prove to more difficult in terms of finding their volumes; some of these objects might include a baseball bat, door knob, or doughnut. We will be looking at finding the volume of an egg, which is similar to a sphere but has more of an irregular oval or spheroid shape—in that it is 'fatter' at the bottom—but first a discusion on the method of volumes by cylindrical shells. The method of volumes by cylindrical shells takes an object of revolution and breaks in down into concentric cylinders of constant t...

Shells and washers can both be used to measure the volume of a region. Volume is used on a 3-dimensional object. To change a 2-dimensional figure to a 3-dimensional figure we revolve the functions we have around the y-axis or the x-axis. To show this I will be using the shells method with the functions \[y=x^3-2x\] and \[y=-x^2\] I will be revolving the region between \[0 ≤ x ≤ 1\] about the y-axis. The functions are graphed in the image below. Using the shells method we would make rectangles parallel to the axis of rotation. For this example we are using the y-axis so the rectangles would be vertical. The more rectangles we make the better the approximation would be for the volume of the object. Revolving these rectangles around the y-axis would make a shell like object. I have made 10 rectangles and revolved it around the y-axis which is shown in the image below. In the image we can see that t...

When calculating the area under a curve, it's easy to split the curve up into slices. These slices are roughly rectangular, and the approximate area can be found by the summation of length times height for each rectangle. If there are two curves, the area between them is simply the area under the top curve minus the area under the bottom curve. This works for two dimensions in the x- and y-planes, but it gets more complicated if these curves get wrapped around an axis and into three dimensions (the z-plane). It is best to visualize the area between two curves in three dimensions as washers or shells. Take, for example, the curves \[y=25-(x-5)^2 \] and \[ y=(x-5)^2-25 \] wrapped around the y-axis. As seen in the picture, this is a good function to teach with because each of the ten layers on the 3D model coorespond to a whole number on the graph. It's easier for beginners to visualize whole numbers rather than fractions. The area of this solid of r...

The Washer Method      When calculating the area under the curve, it may be easier to visualize splitting the curve into multiple rectangles. Then finding the area of each of those rectangles to sum them together for the total area. However, once these curves rotate around an axis, they become a 3D shape and the same struggle to find the volume occurs. Similarly, to calculate the volume of this type of solid, it may be easier to break it down into multiple flat or standardized disks (or if there is a hole they are known as washers). Then finding the volume of each disk (or washer) and sum together for the total volume.  This would be known as the washer method.  Choosing a Basic Function To best understand this concept, I chose a basic function I chose this function because it is important to completely understand how a method works on the most standard equation before moving onto more complicated functions. Even though this is not the best example because i...

In chemistry, all properties of a molecule arise from how the electrons within the molecule interact with the environment around them. In the simplest case, this can lead to a molecule giving up, or taking on an electron to participate in an oxidation-reduction reaction; the basic process that allows for cellular metabolism. In the most complicated case, in organometallic based light-emitting diodes, an electron within the organic part of the molecule can violate quantum rules to flip their spin thanks to the metallic center to form a forbidden triplet state, and then release their energy through phosphorescence to create light.   However, chemists could not immediately understand these phenomena without creating a fundamental model of how an arbitrary particle behaved in an enclosed box, and this lead to the aptly named "particle in a box" model. In such a box, an observer can only determine the velocity or the position of the particle at a given moment, bu...

  Introduction Imagine for a moment that you work for a chocolate factory and help to create new treats. You’re very good at your job and have developed a design for a new candy consisting of a chocolate shell with a caramel-filled cavity. You want the outside of the shell to be 1.5 inches tall and have the same curve as the equation \( f(x)=- \frac{1}{9}\ x^2+ \frac{3}{4}\ \). You also need there to be an opening in the center for the filling, which you want to have the same curve as \( g(x)= \frac{1}{2}\ - \frac{1}{2}\ e^{-x} \). There’s only one problem: before you can begin manufacturing your masterpiece, your boss wants to know what the volume of this candy is so he can order the correct amount of chocolate. Is there a way to calculate the volume of such a complicated shape? Fortunately, there is, with a little help from calculus. What is the Washer Method? As discussed above, we know the wall of our solid will consist of the space between the following t...