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...

...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...

...
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...

...is strong, powerful, dangerous 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...

...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...

...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...

...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)...

...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...

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