Lecture 15 The Definite Integral and Area Under a Curve Definite Integral ---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 definite integral [pic] is the total net change of the antiderivative F over the interval from [pic]

• Properties of Definite Integrals (all of these follow from the FTC) 1. [pic]4. [pic]
2. [pic]5. [pic]
3. [pic], k is a constant.

The method of substitution and the method of integration by parts can also be used to evaluate a definite integral.

[pic]
Examples
5. Find [pic] 6.Find [pic]
7.Find [pic]8.Find [pic]
Area and Integration

There is a connection between definite integrals 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 definite integral [pic]. [pic],

where [pic]is any antiderivative of [pic].

• Why the Integral Formula for Area Works?

[pic]

Let A(x) denote the area of the region under f between a and x, then

[pic]

In general,

[pic]
[pic]
By the definition of the derivative,

[pic]
[pic]
[pic]
Since A(a) is the area under the curve between x = a and x = a, which is clearly zero. Hence, [pic]= the area under f between a and b.

Note:The fundamental theorem requires that the function [pic] is non-negative over the interval [pic]. If [pic]is negative over the interval [pic], the definite integral, [pic], results in a value that is the negative of the area between [pic]and the x-axis from[pic] In such a case, the area between the x-axis and the curve is the absolute value of the definite integral, [pic]....

...DefiniteIntegrals
Section 5.2
OBJECTIVES: - be able to express the areaunder a curve as a definiteintegral and as a limit of Riemann sums
- be able to compute the areaunder 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...

...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 is also known...

...Martyna Wiacek
MTH 116 C- Applied Calculus
11/6/2012
Chapter 5 Writing Assignment
There is a correlation between area, accumulated change, and the definiteintegral 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 definiteintegral 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 definiteintegral 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...

...and not to be trifled with.
On a Monday morning, I deviated from my usual path of taking the bridge over the Gizri area and instead explored the lower (the irony) road. I found myself sitting in a corner dhabba sipping my unhygienic yet delicious pink tea as I re-evaluated the areaunder the bridge: A desolate run down area, filled with potential. It smelt of gloom and despair. You could feel the uneasiness of the people, you could feel their unhappiness. The sun warmed up the stuffy, stinking air which smelled of sweat and rotting fish. To subdue the amalgamated smells, which ruled over the cramped air, was a challenge for even the best air fresheners.
As dawn approached, vans filled with tune deaf children, blind men and women with sleeping infants arrived. The insolent children hid away their shoes after getting off the van, the women took out their children’s medical bills and the supposed blind men put on their glasses and held up their signs. Street begging is a bomb which explodes every day, yet gets bigger. They strategically dispersed covering every nook and cranny. Tugging the dupattas of women; flaunting their bills into cars and ‘accidently’ almost coming in the path of the cars. These were what mornings under the bridge looked like: a well computed elaborate heist.
The street vendors started to assemble their stalls. Each vendor has a designated area where they labored till...

...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 marketing is called word-of-mouth marketing, which relies on the added credibility of person-to-person communication, a personal recommendation.[2]
Career Guidance:
From Institute of Career Guidance
'Career guidance refers to services and activities intended to assist individuals of any age and at any point throughout their lives, to make educational, training and occupational choices and to manage their careers. Such services may be found in schools, universities and colleges, in training institutions, in public employment services, in the workplace, in the voluntary or community sector and in the private sector. The activities may take place on an individual or group basis and may be face-face or at a...

...With over 4,000 employees, and a million patients Peachtree’s IT infrastructure has not kept up with their growth. Peachtree’s reputation is built upon its “quality, consistency, and continuity of care across the entire network” and while doing so they strive for the “highest levels of efficacy, economy, and respect for patients and staff” (Glaser p.1-2).
While the healthcare industry has moved towards standardization Peachtree has resisted. Peachtree’s CEO believes that “the last word in all matters of patient care should rest with the doctor and the patient”, and instead of blanked standardization, he prefers selected standardization of “clinical treatment – immunizations, pharmacy record keeping, aspects of diabetes care” and other areas where there was little disagreement on the best solution.
Peachtree Healthcare will be impacted by the new IT system, it is unavoidable and necessary. In order to make an educated decision on the best IT system for Peachtree these impacts must be analyzed. The inaction of a new IT system will inevitably impact the doctors, patients, administrative staff and possibly the business processes. By forecasting the impacts of a monolithic and SOA based system the best choice can be made.
One of Peachtree’s main concerns is the doctor’s resistance to standardization and a new IT system. Max, the CEO, voices these concerns in Glaser’s article saying “a monolithic system would render the surgical approach difficult to the...

...The Math area is an integral part of the overall Montessori curriculum. Math is all around us. Children are exposed to math in various ways since their birth. They begin to see numbers all around their environment. It is inherent for them to ask questions about time, money and questions about quantities. Math should be included in the Montessori curriculum because math materials are bright, colorful and aesthetically pleasing, math materials are clear and concrete that children are able to understand. For example, children relate numbers with real objects that eventually become abstract ideas, many of the math materials teach different skills at the same time and children are able to work independently and are able to be successful.
Materials in math are colorful, bright. In my classroom, children are drawn to the golden beads especially the one thousand cube, the red and blue rods, the bead cabinets. The smooth texture of the golden, shiny beads are so inviting to children. In my classroom, we often change objects and counters to reflect the theme or seasons throughout the year. For example, during winter we use small snowflakes as counters. We have also used pumpkins during the fall season. We have used shamrocks in the spring.
Math materials use concrete objects to teach abstract ideas like counting. The concrete is the number or quantity. And, the abstract is the numeral or the symbol. During the presentation of the...

...Performance Task in GEOMETRY
* Computation of the surface area, amount and type of needed material and the volume of the package.
Volume
V= L x H x W
= (23 cm) (4 cm) (12cm)
= (276) (4)
= 1 104 cm
Area
A= L x W
= (23cm) (12cm)
= 276cm
Surface Area
A= 2(Lh) + 2(Lw) + 2(Wh) / 2( lh + lw + wh)
= 2(23*4) + 2(23*12) + 2(12*4)
= 2(92) + 2(276) + 2(48)
= 184 + 552 + 96
= 832 cm
* Comparison between the values of surface area and the amount of the needed material.
The total value of the surface area is 832. It is the content of the whole box which is the Borro’s cookies. If we are going to be specific, the total amount of the product is P150.00. Comparing the value of the surface area to the amount of the material, if the surface area is big the quantity of the cookies inside the box have also large amount of the product. What we paid is the cost of the container and the content of the of the Borro’s cookies. The bigger the size of the container, the bigger the content of the product.
* Specification of the packaging material and the costing.
As I made the product from creating Borro’s cookies, it was not based from the real content but it was based to the exact quality, quantity and the looks of my...

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