There is a correlation between area, accumulated change, and the definite integral that we have focused on throughout Chapter 5 in Applied Calculus.
When looking at one rate-of-change function, the accumulated change over an interval and the definite integral are equivalent, their values could be positive, negative or zero. However, the area could never be negative because area is always positive by definition. The accumulated change looks at the whole area of the function that is between the graph and the horizontal axis. For instance, if f (x) is a rate-of-change function the area between f (x) and the x-axis represents the accumulated change between x = a and x = b. However, the definite integral puts specific limits into the function and the area of a particular region can be determined. For example, if f (x) is a rate-of-change function it means that: is what you can consider the area. The accumulation of change in a certain function can be evaluated by using the area of the region between the rate-of-change curve and the horizontal axis.

We also see a similar relationship between the rate-of-change graph and the accumulated graph that we saw in derivatives. A minimum in the accumulated graph is caused by the rate-of-change function crossing over from positive to negative. A maximum in the accumulated graph is a result of the rate-of-change function moving from negative to positive. When there is a maximum or minimum in the rate-of-change graph you get an inflection point in the accumulation graph as well. Also, we see that if the rate-of-change function is negative then the accumulated graph is negative and so the accumulation graph is decreasing. However, when the rate-of-change graph is increasing, it does not affect whether or not the accumulated graph is increasing or decreasing.

There are several problems in our book that demonstrate this...

...DefiniteIntegrals
Section 5.2
OBJECTIVES: - be able to express the area under a curve as a definiteintegral and as a limit of Riemann sums
- be able to compute the area under a curve using a numerical integration procedure
- be able to make a connection with the definition of integration with the limit of a Riemann Sum
Sigma notation enables us to express a large sum in compact form:
[pic]
The Greek capital letter [pic](sigma) stands for “sum.” The index k tells us where to begin the sum (at the number below the [pic]) and where to end (the number above). If the symbol [pic] appears above the [pic], it indicates that the terms go on indefinitely. [pic] is called the norm of the partition which is the biggest [pic] (interval)
Riemann Sum: A sum of the form [pic] where f is a continuous function on a closed interval [a, b]; [pic] is some point in, and [pic] the length of, the kth subinterval in some partition of [a, b].
Big Ideas of a Riemann Sum:
- the limit of a Riemann sum equals the definiteintegral
- rectangles approximate the region between the x-axis
and graph of the function
- A function and an interval are given, the interval is
partitioned, and the height of each rectangle can be
a value at any point in the subinterval
Negative area?
Because the function is not positive, a Riemann sum...

...Before we can discuss both definite and indefinite integrals one must have sufficient and perfect understanding of the word integral or integration. So the questions that arise from this will be “what is integral or integration?”, “why do we need to know or study integral or integration?” and if we understand its concept then “what are its purposes’? These questions should be answered clearly to give a clear, precise meaning and explanation to definite and indefinite integrals.
To answer the first question in a very plain language, integration is simply the reveres of differentiation. And differentiation is, briefly, the measurement of rate of change between two variables, for example, x and y. This mathematical method can be used to reverse derivative back to its original form. For some one that is familiar with derivative, we know that d/dx (x2) = 2x or in mathematical notation we can write it as f ’(x2) = 2x. This is calculated simply by using the derivative formula nxn-1 where x2 will be 2* x2-1 = 2x.
Now to reverse this derivative we have to use law of integral (power rule) that states for f(x), x = xn+1n+1 (normally written as xn+1n+1 + k) now f(x) = 2x will now be equal to 2 * x1+11+1 = 2* x22 + c = x2
This method of reversing the derivative of a function f back to its original form is what is meant by integral. It...

...Lecture 15 The DefiniteIntegral and Area Under a Curve
DefiniteIntegral ---The Fundamental Theorem of Calculus (FTC)
Given that the function [pic] is continuous on the interval [pic] Then,
[pic]
where F could be any antiderivative of f on a ( x ( b. In other words, the definiteintegral [pic] is the total net change of the antiderivative F over the interval from [pic]
• Properties of DefiniteIntegrals (all of these follow from the FTC)
1. [pic] 4. [pic]
2. [pic] 5. [pic]
3. [pic], k is a constant.
Examples
1. Find [pic] 2. Find [pic]
3. Suppose [pic]. Find [pic], hence find [pic]
4. Suppose [pic]. Find.[pic].
• Evaluate DefiniteIntegrals by Substitution
The method of substitution and the method of integration by parts can also be used to evaluate a definiteintegral.
[pic]
Examples
5. Find [pic] 6. Find [pic]
7. Find [pic] 8. Find [pic]
Area and Integration
There is a connection between definiteintegrals and the geometric concept of area. If f(x) is continuous and nonnegative on the interval [pic], then the region A under the graph of f between [pic]has area equal to the definiteintegral [pic].
[pic],
where [pic]is any antiderivative of [pic].
•...

...Student:
1. A person engaged in study; one who is devoted to learning; a learner; a pupil; a scholar; especially, one who attends a school, or who seeks knowledge from professional teachers or from books; as, the students of an academy, a college, or a university; a medical student; a hard student.
2. One who studies or examines in any manner; an attentive and systematic observer; as, a student of human nature, or of physical nature.
Read more at http://www.brainyquote.com/words/st/student224972.html#8e3V1akysIFQymGV.99
Word of Mouth:
From Wikipedia, the free encyclopedia
Word of mouth, or viva voce, is the passing of information from person to person by oral communication, which could be as simple as telling someone the time of day. Storytelling is a common form of word-of-mouth communication where one person tells others a story about a real event or something made up. Oral tradition is cultural material and traditions transmitted by word of mouth through successive generations. Storytelling and oral tradition are forms of word of mouth that play important roles in folklore and mythology. Another example of oral communication isoral history—the recording, preservation and interpretation of historical information, based on the personal experiences and opinions of the speaker. Oral history preservation is the field that deals with the care and upkeep of oral history materials collected by word of mouth, whatever format they may be in. An important area of...

...N.E.D University of Engg. & Tech. CS-14
Integral Calculus:
Definition:
“The branch of mathematics that deals with integrals, especially the methods of
ascertaining indefinite integrals and applying them to the solution of differential
equations and the determining of areas, volumes, and lengths.”
History of Integral Calculus:
Pre-calculus integration:
The first documented systematic technique capable of determiningintegrals is
the method of exhaustion of the ancient Greek astronomer Eudoxus (ca. 370 BC),
which sought to find areas and volumes by breaking them up into an infinite number
of shapes for which the area or volume was known. This method was further
developed and employed by Archimedes in the 3rd century BC and used to calculate
areas for parabolas and an approximation to the area of a circle. Similar methods
were independently developed in China around the 3rd century AD by Liu Hui, who
used it to find the area of the circle. This method was later used in the 5th century by
Chinese father-and-son mathematicians Zu Chongzhi and Zu Geng to find the
volume of a sphere (Shea 2007; Katz 2004, pp. 125–126).
The next significant advances in integral calculus did not begin to appear until the
16th century. At this time the work of Cavalieri with his method of indivisibles, and
work by Fermat, began to lay the foundations of modern calculus, with Cavalieri
computing the...

...Debate Resolution 3:
Be it resolved that development is characterized by stability rather than change.
Introduction
* What is development? Herr (2008) posits that development refers to change or growth that occurs in children.
* ‘The issue of stability versus change relates to whether or not personality traits during infancy endure in children throughout their Life Span’ (Education.com, 2013).
* What is personality? Research shows that personality encompasses a number of characteristics that arise from within an individual. Personality psychology looks at the patterns of thoughts, feelings, and behavior that make a person unique. While, Your Dictionary.com (2013) defines personality traits as actions, attitudes and behaviours you possess.
Summary of Key Arguments
Question: Does personality become stabilized during the first five years of life, as suggested by Sigmund Freud? Reflection: As the twig is bent, so the tree grows. (Clarizio, Craig & Mehrens, (1974).
Answer: On the whole, a child’s personality continues to develop in the direction it started, whether it be a shy, withdrawn girl or an aggressive demanding boy, these characteristics are likely to persist into adulthood…(Clarizio, et al (1974).
Additionally, Windows of Opportunity is a specific time span for normal development of certain types of skills. “Timing is important” Herr (2008). For example, since the critical period for emotional...

...ChangeVs. Development
MGT
Due: 8/21/2003
The concepts of change and development come up frequently in the fields of business, technology, education, sociology, psychology, and many other fields. These concepts may appear to be the same, or similar, but they are very different concepts.
According to Webster's Universal College Dictionary, the definition of change is as follows: "To make different in form; to transform; to exchange for another or others; to give and take reciprocally; to transfer from one to another; to give or get smaller money; to give or get foreign money in exchange for; to remove and replace the coverings or garments of; to become different; to become altered or modified; to become transformed; to transfer between conveyances; to make an exchange; to pass from one phase to another; a replacement or substitution; a transformation or modification; variety or novelty." The synonyms for the word change, as listed in Roget's Desk Thesaurus, are: "alter, modify, make different, adjust, shift, vary, recast, restyle, remodel, reorganize, reform, revolutionize, transfer, transmute, mutate, transform, turn, convert, metamorphose; exchange, replace, substitute, swap, trade, switch, shift, interchange, shuffle, remove and replace; difference, modification, switch, shift, variation, deviation, variety, fluctuation, veering, alteration, conversion, substitution, swapping, reform,...

...except change," said philosopher Heraclitus. Others have called change or variety as 'the spice of life'. So, changes (shuffle or reshuffle) in the government from time to time should come as no surprise to anyone, though changes in the political arena are often viewed with suspicion.
Change is in the very nature of being. Every new day is different from the previous day. Body metabolism is one such process as also growth of trees and revolving of planets. Tides come and go. Sometimes a whole river changes its course as was the case with the Saraswati.
The great insight of the enlightened, Gautam the Buddha, was the everything that is, will change and the changed will change further. Hence, one must neither get attached to joy (happiness) because that will pass away; nor get depressed with sorrow (suffering) because that too will pass away. Nothing is really permanent in this world.
Changes can be categorized under two main types. Changes that take place in nature we have little or no control over. We cannot, for instance, switch the time of tides, which anyway, wait for no one. The other kind of change is the one we witness either in political, social or other fields including the area of personal life. These are changes over which one can exercise some degree of control, changes...

474 Words |
2 Pages

Share this Document

{"hostname":"studymode.com","essaysImgCdnUrl":"\/\/images-study.netdna-ssl.com\/pi\/","useDefaultThumbs":true,"defaultThumbImgs":["\/\/stm-study.netdna-ssl.com\/stm\/images\/placeholders\/default_paper_1.png","\/\/stm-study.netdna-ssl.com\/stm\/images\/placeholders\/default_paper_2.png","\/\/stm-study.netdna-ssl.com\/stm\/images\/placeholders\/default_paper_3.png","\/\/stm-study.netdna-ssl.com\/stm\/images\/placeholders\/default_paper_4.png","\/\/stm-study.netdna-ssl.com\/stm\/images\/placeholders\/default_paper_5.png"],"thumb_default_size":"160x220","thumb_ac_size":"80x110","isPayOrJoin":false,"essayUpload":false,"site_id":1,"autoComplete":false,"isPremiumCountry":false,"userCountryCode":"US","logPixelPath":"\/\/www.smhpix.com\/pixel.gif","tracking_url":"\/\/www.smhpix.com\/pixel.gif","cookies":{"unlimitedBanner":"off"},"essay":{"essayId":36516817,"categoryName":"Periodicals","categoryParentId":"17","currentPage":1,"format":"text","pageMeta":{"text":{"startPage":1,"endPage":2,"pageRange":"1-2","totalPages":2}},"access":"premium","title":"Accumulated Change vs. Definite Integral","additionalIds":[16,7,184,93],"additional":["Law","Education","Law\/Intellectual Property","Education\/Greek System"],"loadedPages":{"html":[],"text":[1,2]}},"user":null,"canonicalUrl":"http:\/\/www.studymode.com\/essays\/Accumulated-Change-Vs-Definite-Integral-1247606.html","pagesPerLoad":50,"userType":"member_guest","ct":10,"ndocs":"1,500,000","pdocs":"6,000","cc":"10_PERCENT_1MO_AND_6MO","signUpUrl":"https:\/\/www.studymode.com\/signup\/","joinUrl":"https:\/\/www.studymode.com\/join","payPlanUrl":"\/checkout\/pay","upgradeUrl":"\/checkout\/upgrade","freeTrialUrl":"https:\/\/www.studymode.com\/signup\/?redirectUrl=https%3A%2F%2Fwww.studymode.com%2Fcheckout%2Fpay%2Ffree-trial\u0026bypassPaymentPage=1","showModal":"get-access","showModalUrl":"https:\/\/www.studymode.com\/signup\/?redirectUrl=https%3A%2F%2Fwww.studymode.com%2Fjoin","joinFreeUrl":"\/essays\/?newuser=1","siteId":1,"facebook":{"clientId":"306058689489023","version":"v2.8","language":"en_US"},"analytics":{"googleId":"UA-32718321-1"}}