Integration Techniques

Group MembersSam Taylor

Patience CantyAustin Hood

Definition of an Integral• - inverse of differentiation – Uses• areas under curved surfaces• centres of mass• Volumes of solids

• Formulas for Integration

Indefinite Definite

World Applications of Integration

The Petronas Towers in Kuala Lumpur experience high forces due to winds. Integration was used to design the

building for strength

Historically, one of the first uses of integration was in finding the volumes of wine-casks (which have a curved surface)

The Sydney Opera House is a very unusual design based on slices out of a ball. Many differential

equations (one type of integration) were solved in the design of this building

http://www.intmath.com/Integration/Integration-intro.php

http://www.intmath.com/Integration/Integration-intro.php

http://www.intmath.com/Integration/Integration-intro.php

World Applications of Integration cont.The Head Injury Criterion (HIC)

The head Injury Criterion (HIC) was developed, it is based on the average value of the acceleration over the most critical part of the deceleration. It

more accurately describes the likelihood of certain injuries in a crash

The average value of the acceleration a(t) over the time interval t1 to t2 is given by

For the HIC, this was modified (based on experimental data) as follows:

The formula means: The HIC is the maximum value over the critical time period t sub 1 to t sub2 for the expression in {}. The index 2.5 is chosen for the head, based on experiments.

http://www.intmath.com/Applications-integration/HIC5.php

History of Integration

Over 2000 years ago, Archimedes (287-212 BC) found formulas for the surface areas and volumes of solids such as the sphere, the cone, and the

paraboloid. His method of integration was remarkably modern considering that he did not have algebra, the function concept, or even

the decimal representation of numbers.

http://www.fredsakademiet.dk/library/science/science3.htm

Leibniz (1646-1716) and Newton (1642-1727) independently discovered calculus. Their key idea was that differentiation and integration undo each other. Using this symbolic connection, they were able to solve an enormous number of important

problems in mathematics, physics, and astronomy.

http://www.gwleibniz.com/britannica_pages/leibniz/leibniz_gif.html http://inversesquare.wordpress.com/2007/12/14/friday-isaac-newton-blogging-an-apple-tree-of-knowledge/

History of Integration cont.

http://www.math.hope.edu/newsletter/2006-07/05-13.html

Gauss (1777-1855) made the first table of integrals, and with many others

continued to apply integrals in the mathematical and physical sciences.

Cauchy (1789-1857) took integrals to the complex domain. Riemann (1826-1866) and Lebesgue (1875-1941) put definite

integration on a firm logical foundation.

http://www.mlahanas.de/Physics/Bios/RelativityMathematicians.html

In the 20th century before computers, mathematicians developed the theory of integration and applied it to write tables of integrals and

integral transforms.

http://www5.in.tum.de/lehre/seminare/math_nszeit/SS03/vortraege/frank/

History of Integration cont.In 1969 Risch made the major breakthrough in algorithmic

indefinite integration when he published his work on the general theory and practice of integrating elementary

functions.

The capability for definite integration gained substantial power in Mathematica, first released in 1988. Comprehensiveness and

accuracy have been given strong consideration in the development of Mathematica and have been successfully accomplished in its

integration code

http://www.ebookee.net/The-Mathematica-Book-Fifth-Edition_37396.html

POWER RULE1

1

nn u

u du Cn

1n

where

C = Constant of integrationu = Functionn = Power du = Derivative

=

u = xdu = dxn = 2

=

Examples1.

u = x du = dx n = 2

u = x du = dx n = 1

u = x du = dx n = o

Solution:

2.

u = 2x + 1du = 2dxn = 15

Solution:

Trigonometric Integration and Natural Log Integration Formulas

Example of Trigonometric

Solution:

Example 2 of Trigonometric

OR

Solution: Solution:

Example of Natural Log u = 2x + 3du = 2dx n = -1

Solution:

Example 2 of Natural Log

Solution:

U - Substitution

When to use U-Sub: The problem must be

two algebraic functions One of them is NOT the derivative of the other

Examples of this would include:

Examples

1.

u =

Solution:

2.

Solution:

Integrating Powers of Sine and Cosine

Rules and Ways to integrate: Integrating Odd Powers

Integrating Odd and Even Powers Integrating Even Powers

Integrating Odd Powers

Solution:

Integrating Odd and Even Powers

Solution:

Integrating Even Powers

Half – Angle Formulas

Solution:

Integration by PartsIf the functions are not related then use integration by parts

Special things to Consider: Use lnx as the u variable . u = ln(x)

Example

Solution:

Important/Unusual Integrals

+C

+C

+C

+C

Example 2

Solution:

Integration by Partial FractionsMust Haves:

Expressions must be polynomialsPower Rule must be used at some pointDenominator is factorable, then partial

factorsPower or exponent = how many

variables or fractionsExample 1

Solution:

Example 2

Solution:

Definite IntegrationUsed when the

numerical bounds of the object are known

First Fundamental Theorem of Calculushttp://www.sosmath.com/calculus/integ/integ01/integ01.html

Examples of Definite Problems

Plug 2 inPlug 0 in

Solution:

Solution:

© Patience Canty, Austin Hood, and Sam Taylor Feb. 17 2010

Bibliographywww.awesomebackgrounds.com

www.iphotostock.com

http://www.intmath.com/Integration/Integration-intro.php

http://www.math.hope.edu/newsletter/2006-07/05-13.html

http://www.mlahanas.de/Physics/Bios/RelativityMathematicians.html

http://www5.in.tum.de/lehre/seminare/math_nszeit/SS03/vortraege/frank/

http://www.gwleibniz.com/britannica_pages/leibniz/leibniz_gif.html

http://inversesquare.wordpress.com/2007/12/14/friday-isaac-newton-blogging-an-apple-tree-of-knowledge/

http://www.fredsakademiet.dk/library/science/science3.htm

http://www.intmath.com/Applications-integration/HIC5.php

http://integrals.wolfram.com/about/history/

http://www.sosmath.com/calculus/integ/integ01/integ01.html