Applications of Multiple Integrals in Engineering

Applications of Multiple Integrals in Engineering Integrals are used to consider the full body on the basis of analysis done on a small part of it, but these analysis are just on a single dimension of any body, for e.g. if we take a cuboid it has three dimensions i.e. its length, breadth and the height. But by to analyse it we have consider all of its dimensions, this is where Multiple Integrals into application. Multiple integrals are there for multiple dimensions of a body. Now for taking a cuboid into consideration we need to be working in Triple Integration. The definition of a definite integrals for functions of single variable, while working with the integral of single variable is as below, f(x) dx we think of xs as coming from the interval a ≠¤ x ≠¤ b . For these integrals we can say that we are integrating over the interval a ≠¤ x ≠¤ b . Note that this does assume that a < b , however, if we have b < a then we can just use the interval b ≠¤ x ≠¤ a . Now, when we derived the definition of the definite integral we first thought of this as an area problem. We first asked what the area under the curve was and to do this we broke up the interval a ≠¤ x ≠¤ b into n subintervals of width Δx and choose a point, , from each interval as shown below, Each of the rectangles has height of f() and we could then use the area of each of these rectangles to approximate the area as follows, A ≈ f(Δx + f(Δx + f() Δx f() Δx To get the exact area then we take the limits as n that go till infinity, to fulfil the definition of definite integrals = This was how we integrate for a single dimension in a single variable, but we want to integrate a function of two variables, f (x, y). With functions of one variable we integrated over an interval (i.e. a one-dimensional space) and so it makes some sense then that when integrating a function of two variables we will integrate over a region of (two dimensional space). DOUBLE INTEGRALS We will start out by assuming that the region in is a rectangle which we will denote as follows, R = [a,b]*[c, d] This means that the ranges for x and y are a ≠¤ x ≠¤ b and c ≠¤ y ≠¤ d . Also, we will initially assume that f ( x, y) ≠¥ 0 although this doesnt really have to be the case. Lets start out with the graph of the surface S give by graphing f (x, y) over the rectangle R. Now, just like with functions of one variable lets first talk about what the volume of the region under S (and above the xy-plane) is. We will first approximate the volume much as we approximated the area above. We will first divide up a ≠¤ x ≠¤ b into n subintervals and divide up c ≠¤ y ≠¤ d into m subintervals. This will divide up R into a series of smaller rectangles and from each of these we will choose a point (,). Here is a sketch of this set up. Now, over each of these smaller rectangles we will construct a box whose height is given by f (,). Here is a sketch of that. Each of the rectangles has a base area of ΔA and a height of f (,) so the volume of each of these boxes is f (,).ΔA. The volume under the surface S is then approximately, V ≈ [f (,).ΔA] We will have a double sum since we will need to add up volumes in both the x and y directions. To get a better estimation of the volume we will take n and m larger and larger and to get the exact volume we will need to take the limit as both n and m go to infinity. In other words, V = [f (,).ΔA] Now, this should look familiar. This looks a lot like the definition of the integral of a function of single variable. In fact this is also the definition of a double integral, or more exactly an integral of a function of two variables over a rectangle. Here is the official definition of a double integral of a function of two variables over a rectangular region R as well as the notation that well use for it. = [f (,).ΔA] Note the similarities and differences in the notation to single integrals. We have two integrals to denote the fact that we are dealing with a two dimensional region and we have a differential here as well. Note that the differential is dA instead of the dx and dy that were used to seeing. Note as well that we dont have limits on the integrals in this notation. Instead we have the R written below the two integrals to denote the region that we are integrating over. Note that one interpretation of the double integral of f (x, y) over the rectangle R is the volume under the function f (x, y) (and above the xy-plane). V = We can use this double sum in the definition to estimate the value of a double integral if we need to. We can do this by choosing (,) to be the midpoint of each rectangle. When we do this we usually denote the point as (,). This leads to the Midpoint Rule, ≈ [f ( , ).ΔA] Now we be looking at how to actually compute double integrals. Fubinis Theorem If f (x, y) is continuous on R = [a,b]*[c, d] then, = These integrals are called iterated integrals. Note that there are in fact two ways of computing a double integral and also notice that the inner differential matches up with the limits on the inner integral and similarly for the out differential and limits. In other words, if the inner differential is dy then the limits on the inner integral must be y limits of integration and if the outer differential is dy then the limits on the outer integral must be y limits of integration. Now, on some level this is just notation and doesnt really tell us how to compute the double integral. Lets just take the first possibility above and change the notation a little. We will compute the double integral by first computing and we compute this by holding x constant and integrating with respect to y as if this were an single integral. This will give a function involving only xs which we can in turn integrate. Weve done a similar process with partial derivatives. To take the derivative of a function with respect to y we treated the xs as constants and differentiated with respect to y as if it was a function of a single variable. Double integrals work in the same manner. We think of all the xs as constants and integrate with respect to y or we think of all ys as constants and integrate with respect to x. In this case we will integrate with respect to y first. So, the iterated integral that we need to compute is, When setting these up make sure the limits match up to the differentials. Since the dy is the inner differential (i.e. we are integrating with respect to y first) the inner integral needs to have y limits. To compute this we will do the inner integral first and we typically keep the outer integral around as follows, (2x) dx = dx = dx Remember that we treat the x as a constant when doing the first integral and we dont do any integration with it yet. Now, we have a normal single integral so lets finish the integral by computing this. = 7 = 84 we can do the integral in either direction. However, sometimes one direction of integration is significantly easier than the other so make sure that you think about which one you should do first before actually doing the integral. The next topic of this section is a quick fact that can be used to make some iterated integrals somewhat easier to compute on occasion. Fact If f (x, y) = g (x)h( y) and we are integrating over the rectangle R = [a,b]*[c, d] then, dA = ().() So, if we can break up the function into a function only of x times a function of y then we can do the two integrals individually and multiply them together. Lets do a example using this integral. Example- Evaluate , R= [-2,3]*[0,] Solution- Since the integrand is a function of x times a function of y we can use the fact. ().() = . (] = + [y+] = DOUBLE INTEGRALS OVER GENERAL REGIONS In the previous section we looked at double integrals over rectangular regions. The problem with this is that most of the regions are not rectangular so we need to now look at the following double integral à ²Ãƒ ² f(x,y) dA Such figures show us the area that we have to consider and the details about that area i.e. the points from which its starting or finishing at etc. Lets do an example to understand these type of figures and problems well, Example- dA, where D is the region bounded by y = y = . Solution- In this case we need to determine the two inequalities for x and y that we need to do the integral. The best way to do this is the graph the two curves. Here is a sketch. So, from the sketch we can see that that two inequalities are, 0 ≠¤ x ≠¤ 1 , ≠¤ y ≠¤ √x We can now do the integral, dA = dydx = [2xy ]dx = + = TRIPLE INTEGRAL Now that we know how to integrate over a two-dimensional region we need to move on to integrating over a three-dimensional region. We used a double integral to integrate over a two-dimensional region and so it shouldnt be too surprising that well use a triple integral to integrate over a three dimensional region. The notation for the general triple integrals is, dV Lets start simple by integrating over the box, B = [a,b]*[c, d]*[r, s] Note that when using this notation we list the xs first, the ys second and the zs third. The triple integral in this case is, = Note that we integrated with respect to x first, then y, and finally z here, but in fact there is no reason to the integrals in this order. There are 6 different possible orders to do the integral in and which order you do the integral in will depend upon the function and the order that you feel will be the easiest. We will get the same answer regardless of the order however. Lets do a example of this type of triple integral. Example-Determine the volume of the region that lies behind the plane x + y + z = 8 and in front of the region in the yz-plane that is bounded by z = √y and z = y . Solution- In this case weve been given D and so we wont have to really work to find that. Here is a sketch of the region D as well as a quick sketch of the plane and the curves defining D projected out past the plane so we can get an idea of what the region were dealing with looks like. Now, the graph of the region above is all okay, but it doesnt really show us what the region is. So, here is a sketch of the region itself. Here are the limits for each of the variables. 0 ≠¤ y ≠¤ 4 ≠¤ z ≠¤ The volume is then, V = = [ = [ = dz =dy =(8-) = So this is it about the triple integrals as well as the multiple integrals, there are yet some details which are not covered like double integrals in polar coordinates, triple integrals in cylindrical coordinates, and triple integrals in spherical coordinates etc. APPLICATIONS IN MECHANICAL ENGINEERING Now the applications of multiple integrals in mechanical engineering are the basic applications of them i.e. to find areas and volumes of various bodies just by taking a little part of them into consideration. And this is applicable in various fields like while preparing a machine,or the parts to fitted in any machine its size and volume etc. are very important.

Leadership Challenge: Selecting Team Members

When we think leader, we think of the person in charge, the person who is making all the decisions within an organization. We believe that a leader should be charismatic, smart and able to influences others to do what they want and be successful at their role. Nahavandi (2006) defines a leader as a person who influences individuals and groups within an organization, helps them in establishing goals and guides them towards achievement of those goals, which allows them to be effective. In order to be a good leader you will need to be effective in your role. Effective leaders are proven when their supporters achieve their goals, can work together as a team and can adjust to the many changes within the company and the changing demands from exterior forces. Is there a difference between a manager and a leader? Most people would say that a good manager can be a good leader. The functions of a manager are similar to that of a leader; the functions of a manager are to bring order and regularity to the company by planning, budgeting, organizing, staffing directing and controlling. Throughout this report we will show the differences between leading and managing. Should the company move towards a team based, empowered organization and why. We will provide ways in which to deal with resistance to the new changes within the company. Leading vs. Managing Leading and managing is said to be different in the way they organize people. A manager’s role is to capitalize on the output of the company through managerial operation by organization, planning, staffing, directing and controlling. These functions allow the manager to focus on the resources in addition to people while leading focuses on influencing people. Both are needed in order for a company to be successful and effective. They are often considered to be the same thing, the main distinction between leading and managing is that we manage things and lead people. Pascale (1990) states that a managers do things right, while leaders do the right thing. You will need to lead the managers into the need of a team-based environment which will allow them to manage the lowest level employee to achieve their goals. The effectiveness of teamwork in a transfer to a team-based organization, consistency improvement practices by showing them learning by leading as an example, mentoring and guided delegation are important to improve the social effectiveness of the team as a whole. Team-based, empowered organization Moving to a team based environment in order to increase customer satisfaction can be considered to be a good thing for the company on a whole. Allowing the lower level employees the ability to make decisions that are within the best interest of the customer and that will benefit the company. They will get a sense of pride in their work because they are more involved in the development and decision making process of the company. There are many companies today who operate under a team-based environment, like 3M, General Electric and Deere & Co. They are known for their good management practice. While other companies were moving toward a more technology environment Deere realized to having a world class company was in the organizing its people. Deere brought their product and manufacturing engineering team together to create the yellow backhoe loader-digger. By doing so Deere the teams shift from tentative synchronized engineering to organized united engineering. Both teams were involved from the design phase to the production phase. In 1988 for the first time Deere had everyone from all the department involve in the production of their new machine, tool design, welding, product and manufacturing engineering, purchasing, the shop floor, and the accounting department. Moving your company to a team-based environment will benefit everyone involve from the lowest level employee to the managers and executive.

The Definitive Strategy to Professional Research Paper Format

The Definitive Strategy to Professional Research Paper Format You'll also wish to consider employing a literature overview. The paper should get off to a good beginning when it comes to clarifying unfamiliar information as not to depart from your readers puzzled. Writing research paper isn't just merely a compilation of related literature that could support the argument you're trying to raise, the findings you're trying to get, or the question you're attempting to reply. A research paper that concerns scientific issues is not only a free-style essay where you are able to tell readers whatever you desire. You will be happy you filled up the purchase form. There's a format for every person to follow for the simplicity of understanding. Even when you have necessary data, it can be pretty tricky to present it in an appropriate systematic manner in line with the rules of formalization. On top of that, you're confident your personal data will be held in strictest confidence. The Benefits of Professional Research Paper Format There it is possible to learn to construct the structure and its elements. A consistent structure ought to be used in the key headings and the subheadings. In the event the structure of the chief body has many levels, it is preferable to use different numbering styles. The precise structure of your paper will vary somewhat depending on the form of paper you've been requested to write. Utilizing APA format means you're following an internationally acceptable writing style to which many of the readers are extremely familiar too. This format is going to be used. The format to be used is dependent upon the topic that you have chosen. The format of your abstract also depends upon the sort of paper you're writing. For your research paper example to get going, you should announce your introduction with few sentences that show what type of research questions you'll be asking. Additionally, failing to acknowledge the work of others m ay lead to accusations of plagiarism, which then can cause consequences like a failing grade or even getting fired from your work. If your report is poorly written, individuals will tend to presume your research is also poor and you're giving the assessor a simple excuse to offer you a lower grade. A whole lot of students feel the impulse to begin writing immediately, although in case you speak to any professor out there, they will inform you how a superior illustration of a research paper is always superseded with a good plan. So, the one thing you need to do is simply to organize information you've found and texts you have written for your research. Let's consider the components of your research program and the way to compose a structured outline that will permit you to begin a research paper as fast and effectively as possible. Great research paper's examples almost always concentrate on novel suggestions to address which will be of value in the area. In case you're not able to do the research paper because of any reason, you can depend on ProfEssays to write for you in accordance to your requirements. Indicate the way the approach fits the general research design. Regarding, from which angle you examine your topic, it is going to reflect distinctive facets. Professional Research Paper Format - the Conspiracy If you're not certain how to then ask the computer department for support. Personal information is another means to receive information. Since you are going to learn from looking through any superior research paper example, writing a fantastic paper involves so far more than simply throwing a whole lot of text and citations into a word processor and hoping for the very best. College students should use a suitable numbering for unique sections. Fantastic research papers are all over the net for one to read. Before you begin your essay or research paper, you should get familiar with some overall APA rules. An individu al should realize that every Research Paper is a sophisticated writing because it must contain distinctive research and distinctive idea. A research paper is quite a significant kind of academic writing. Regardless of the character of your research, if you're writing a paper an outline can help you to not just organize your thoughts, but in addition function as the template for your whole paper. The goal of your abstract is to supply a brief yet thorough summary of your paper. After all, you're making a paper outline so that it is possible to guide yourself towards starting a research paper, so obviously, you won't need to enter the literature review practice. A crucial part of any research paper outline is going to be a literature review.

The Definitive Strategy to Professional Research Paper Format

The Definitive Strategy to Professional Research Paper Format You'll also wish to consider employing a literature overview. The paper should get off to a good beginning when it comes to clarifying unfamiliar information as not to depart from your readers puzzled. Writing research paper isn't just merely a compilation of related literature that could support the argument you're trying to raise, the findings you're trying to get, or the question you're attempting to reply. A research paper that concerns scientific issues is not only a free-style essay where you are able to tell readers whatever you desire. You will be happy you filled up the purchase form. There's a format for every person to follow for the simplicity of understanding. Even when you have necessary data, it can be pretty tricky to present it in an appropriate systematic manner in line with the rules of formalization. On top of that, you're confident your personal data will be held in strictest confidence. The Benefits of Professional Research Paper Format There it is possible to learn to construct the structure and its elements. A consistent structure ought to be used in the key headings and the subheadings. In the event the structure of the chief body has many levels, it is preferable to use different numbering styles. The precise structure of your paper will vary somewhat depending on the form of paper you've been requested to write. Utilizing APA format means you're following an internationally acceptable writing style to which many of the readers are extremely familiar too. This format is going to be used. The format to be used is dependent upon the topic that you have chosen. The format of your abstract also depends upon the sort of paper you're writing. For your research paper example to get going, you should announce your introduction with few sentences that show what type of research questions you'll be asking. Additionally, failing to acknowledge the work of others m ay lead to accusations of plagiarism, which then can cause consequences like a failing grade or even getting fired from your work. If your report is poorly written, individuals will tend to presume your research is also poor and you're giving the assessor a simple excuse to offer you a lower grade. A whole lot of students feel the impulse to begin writing immediately, although in case you speak to any professor out there, they will inform you how a superior illustration of a research paper is always superseded with a good plan. So, the one thing you need to do is simply to organize information you've found and texts you have written for your research. Let's consider the components of your research program and the way to compose a structured outline that will permit you to begin a research paper as fast and effectively as possible. Great research paper's examples almost always concentrate on novel suggestions to address which will be of value in the area. In case you're not able to do the research paper because of any reason, you can depend on ProfEssays to write for you in accordance to your requirements. Indicate the way the approach fits the general research design. Regarding, from which angle you examine your topic, it is going to reflect distinctive facets. Professional Research Paper Format - the Conspiracy If you're not certain how to then ask the computer department for support. Personal information is another means to receive information. Since you are going to learn from looking through any superior research paper example, writing a fantastic paper involves so far more than simply throwing a whole lot of text and citations into a word processor and hoping for the very best. College students should use a suitable numbering for unique sections. Fantastic research papers are all over the net for one to read. Before you begin your essay or research paper, you should get familiar with some overall APA rules. An individu al should realize that every Research Paper is a sophisticated writing because it must contain distinctive research and distinctive idea. A research paper is quite a significant kind of academic writing. Regardless of the character of your research, if you're writing a paper an outline can help you to not just organize your thoughts, but in addition function as the template for your whole paper. The goal of your abstract is to supply a brief yet thorough summary of your paper. After all, you're making a paper outline so that it is possible to guide yourself towards starting a research paper, so obviously, you won't need to enter the literature review practice. A crucial part of any research paper outline is going to be a literature review.

The Definitive Strategy to Professional Research Paper Format

The Definitive Strategy to Professional Research Paper Format You'll also wish to consider employing a literature overview. The paper should get off to a good beginning when it comes to clarifying unfamiliar information as not to depart from your readers puzzled. Writing research paper isn't just merely a compilation of related literature that could support the argument you're trying to raise, the findings you're trying to get, or the question you're attempting to reply. A research paper that concerns scientific issues is not only a free-style essay where you are able to tell readers whatever you desire. You will be happy you filled up the purchase form. There's a format for every person to follow for the simplicity of understanding. Even when you have necessary data, it can be pretty tricky to present it in an appropriate systematic manner in line with the rules of formalization. On top of that, you're confident your personal data will be held in strictest confidence. The Benefits of Professional Research Paper Format There it is possible to learn to construct the structure and its elements. A consistent structure ought to be used in the key headings and the subheadings. In the event the structure of the chief body has many levels, it is preferable to use different numbering styles. The precise structure of your paper will vary somewhat depending on the form of paper you've been requested to write. Utilizing APA format means you're following an internationally acceptable writing style to which many of the readers are extremely familiar too. This format is going to be used. The format to be used is dependent upon the topic that you have chosen. The format of your abstract also depends upon the sort of paper you're writing. For your research paper example to get going, you should announce your introduction with few sentences that show what type of research questions you'll be asking. Additionally, failing to acknowledge the work of others m ay lead to accusations of plagiarism, which then can cause consequences like a failing grade or even getting fired from your work. If your report is poorly written, individuals will tend to presume your research is also poor and you're giving the assessor a simple excuse to offer you a lower grade. A whole lot of students feel the impulse to begin writing immediately, although in case you speak to any professor out there, they will inform you how a superior illustration of a research paper is always superseded with a good plan. So, the one thing you need to do is simply to organize information you've found and texts you have written for your research. Let's consider the components of your research program and the way to compose a structured outline that will permit you to begin a research paper as fast and effectively as possible. Great research paper's examples almost always concentrate on novel suggestions to address which will be of value in the area. In case you're not able to do the research paper because of any reason, you can depend on ProfEssays to write for you in accordance to your requirements. Indicate the way the approach fits the general research design. Regarding, from which angle you examine your topic, it is going to reflect distinctive facets. Professional Research Paper Format - the Conspiracy If you're not certain how to then ask the computer department for support. Personal information is another means to receive information. Since you are going to learn from looking through any superior research paper example, writing a fantastic paper involves so far more than simply throwing a whole lot of text and citations into a word processor and hoping for the very best. College students should use a suitable numbering for unique sections. Fantastic research papers are all over the net for one to read. Before you begin your essay or research paper, you should get familiar with some overall APA rules. An individu al should realize that every Research Paper is a sophisticated writing because it must contain distinctive research and distinctive idea. A research paper is quite a significant kind of academic writing. Regardless of the character of your research, if you're writing a paper an outline can help you to not just organize your thoughts, but in addition function as the template for your whole paper. The goal of your abstract is to supply a brief yet thorough summary of your paper. After all, you're making a paper outline so that it is possible to guide yourself towards starting a research paper, so obviously, you won't need to enter the literature review practice. A crucial part of any research paper outline is going to be a literature review.

Ideas for Thank You Card Quotes

Do you say thank you to your host after attending a party? Do you say thank you to friends who bring you lovely presents? Sometimes mere thanks may sound insincere. Express your heartfelt gratitude by sending thank you cards. Use these quotes to make your cards really special. Quotations of Gratitude Irving BerlinGot no checkbooks, got no banks.Still Id like to express my thanks.I got the sun in the mornin and the moon at night.AnonymousI would thank you from the bottom of my heart, but for you my heart has no bottom.Oscar WildeThe smallest act of kindness is worth more than the grandest intention.Ralph Waldo EmersonFor each new morning with its light,For rest and shelter of the night,For health and food, for love and friends,For everything Thy goodness sends.William ShakespeareI can no other answer make, but, thanks, and thanks.G K ChestertonYou say grace before meals. All right. But I say grace before the concert and the opera, and grace before the play and pantomime, and grace before I open a book, and grace before sketching, painting, swimming, fencing, boxing, walking, playing, dancing and grace before I dip the pen in the ink.James Russell LowellNot what we give,But what we share,For the gift without the giverIs bare.John Greenleaf WhittierNo longer forward nor behindI look in hope or fear;But, grateful, take the good I find,The best of now and here.br/>Helen KellerI thank God for my handicaps for, through them, I have found myself: my work, and my God.Benjamin DisraeliI feel a very unusual sensation -- if it is not indigestion, I think it must be gratitude.George EllistonHow beautiful a day can beWhen kindness touches it!EE CummingsI thank you God I thank you God for this most amazing day, for the leaping greenly spirits of trees, and for the blue dream of sky and for everything which is natural, which is infinite, which is yes.OvidThanks are justly due for boons unbought.Henry Van DykeBe glad of life because it gives you the chance to love, and to work, and to play and to look up at the stars.