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...

...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...

...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 determining...

...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...

...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...

...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...

...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...

...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)...