OBJECTIVES: - be able to express the area under a curve as a definite integral 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 definite integral

- 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
does not represent a sum of areas of the rectangles.
It represents the sum of areas above the x-axis subtract
the sum of areas below the x-axis.
Area Under a Curve (as a Definite Integral)
If [pic] is nonnegative and integrable over a closed interval [a, b], then the area under the curve y = f(x) from a to b is the integral of f from a to b.[pic]

EX1: Evaluate each integral
EX1a: [pic]EX1b: [pic]

EX2: Use the graph of the integrand and areas to evaluate each integral. EX2a: [pic]EX2b: [pic]...

...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...

...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 ofDefiniteIntegrals (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].
• Why the...

...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...

...Integral calculus is the study of mathematical integration dealing with integrals. These integrals, also known as area under the curve, are used to determine the following: areas, volumes, and lengths. There are a multitude of areas of real-world mathematics to which integral calculus is used. Integration is most often applied in physics, biology, and even chemistry.
Integration is applied to physics in many situations. One important situation is finding the moment of inertia. The moment of inertia is the measure of the resistance of a rotating body to a change in motion. To find the moment of inertia, you find the area under, and also between the curve(s).
An application in which integration is key to reaching the target value is a PID Controller. The purpose of a PID Controller is to determine the error between what is measured and what is expected. The “I” or integration part of the controller is the collected past errors. The actual integral is the total of the instantaneous error over time. All parts of the error are included, meaning duration and amount.
Integral mathematics has much to do with the duration and amount of something. One major example is population analysis. Population analysis is a form of integral math that is applied for biology. The births in the present year rely on many things from previous years, most obviously the amount of births in the...

...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 derivatives. A minimum in the accumulated graph is caused by the rate-of-change function crossing over from positive to negative. A...

...AP Calculus Name ____________________________ Period _____
Functions Defined by Integrals
The graph below is, the derivative of. The graph consists of two semicircles and one line segment. Horizontal tangents are located at and and a vertical tangent is located at.
1. On what interval is increasing? Justify your answer.
2. For what values of x does have a relative minimum? Justify.
3. On what intervals is concave up? Justify
4. For what values of x is undefined?
5. Identify the x-coordinates for all point of inflection of . Justify.
6. For what value of x does reach its maximum value on the closed interval [0, 17]? Justify
7. For what value of x does reach its minimum value on the closed interval [0, 17]? Justify
8. If , find .
9. Let . The graph of f is comprised of line segments and a quarter of a circle on [-2, 6].
a) Find and
b) Determine the open intervals where is increasing. Justify your answer.
c) Determine the intervals where is concave down. Justify your answer.
d) Sketch .
e) Determine the absolute maximum and minimum of on [-2, 6].
10. Let , where the graph of f , defined on [-5, 5], is comprised of line segments.
a) Determine the domain of .
b) Determine the range of .
c) Determine the relative extrema. Justify your answer.
11. The graph of consists of a semicircle and two line segments as shown.
a) Determine the values of x on the open interval (-2, 7)...

...Chapter 4
Vector integrals and integral theorems
Last revised: 1 Nov 2010.
Syllabus covered:
1. Line, surface and volume integrals.
2. Vector and scalar forms of Divergence and Stokes’s theorems. Conservative ﬁelds: equivalence to curl-free
and existence of scalar potential. Green’s theorem in the plane.
Calculus I and II covered integrals in one, two and three dimensional Euclidean (ﬂat) space (i.e. R, R2
and R3 ). We are still working in R3 so there is no generalization to be applied to volume or triple integrals,
but we will generalise one dimensional integration from a straight line to an integral along a curve, and we
will generalise two-dimensional integration from a region in a plane to a curved surface.
We will also be working with integration of vectors, though in many cases we will be using a scalar
product so the ﬁnal quantity to be integrated becomes a scalar. In the cases with a scalar product:
f (x) dx generalizes to
C
F · dr on a curve C , called a line integral (section 4.1).
f (x, y) dx dy generalizes to
S
F · dS over a surface S , called a surface integral (section 4.2).
We will then have to study the generalizations of
b
a
df
dx = f (b) − f (a) ,
dx
(4.1)
called the ‘fundamental theorem of calculus’, which we use in the proofs. This theorem relates a onedimensional integral to a (pair...

...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 product. I got the idea by creating a cookie-box because people usually recognize the product from a...