integration techniques group members sam taylor patience canty austin hood
TRANSCRIPT
![Page 1: Integration Techniques Group Members Sam Taylor Patience Canty Austin Hood](https://reader035.vdocuments.us/reader035/viewer/2022081603/56649d9d5503460f94a86b8c/html5/thumbnails/1.jpg)
Integration Techniques
Group MembersSam Taylor
Patience CantyAustin Hood
![Page 2: Integration Techniques Group Members Sam Taylor Patience Canty Austin Hood](https://reader035.vdocuments.us/reader035/viewer/2022081603/56649d9d5503460f94a86b8c/html5/thumbnails/2.jpg)
Definition of an Integral• - inverse of differentiation – Uses• areas under curved surfaces• centres of mass• Volumes of solids
• Formulas for Integration
Indefinite Definite
![Page 3: Integration Techniques Group Members Sam Taylor Patience Canty Austin Hood](https://reader035.vdocuments.us/reader035/viewer/2022081603/56649d9d5503460f94a86b8c/html5/thumbnails/3.jpg)
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
![Page 4: Integration Techniques Group Members Sam Taylor Patience Canty Austin Hood](https://reader035.vdocuments.us/reader035/viewer/2022081603/56649d9d5503460f94a86b8c/html5/thumbnails/4.jpg)
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
![Page 5: Integration Techniques Group Members Sam Taylor Patience Canty Austin Hood](https://reader035.vdocuments.us/reader035/viewer/2022081603/56649d9d5503460f94a86b8c/html5/thumbnails/5.jpg)
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/
![Page 6: Integration Techniques Group Members Sam Taylor Patience Canty Austin Hood](https://reader035.vdocuments.us/reader035/viewer/2022081603/56649d9d5503460f94a86b8c/html5/thumbnails/6.jpg)
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/
![Page 7: Integration Techniques Group Members Sam Taylor Patience Canty Austin Hood](https://reader035.vdocuments.us/reader035/viewer/2022081603/56649d9d5503460f94a86b8c/html5/thumbnails/7.jpg)
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
![Page 8: Integration Techniques Group Members Sam Taylor Patience Canty Austin Hood](https://reader035.vdocuments.us/reader035/viewer/2022081603/56649d9d5503460f94a86b8c/html5/thumbnails/8.jpg)
POWER RULE1
1
nn u
u du Cn
1n
where
C = Constant of integrationu = Functionn = Power du = Derivative
=
u = xdu = dxn = 2
=
![Page 9: Integration Techniques Group Members Sam Taylor Patience Canty Austin Hood](https://reader035.vdocuments.us/reader035/viewer/2022081603/56649d9d5503460f94a86b8c/html5/thumbnails/9.jpg)
Examples1.
u = x du = dx n = 2
u = x du = dx n = 1
u = x du = dx n = o
Solution:
![Page 10: Integration Techniques Group Members Sam Taylor Patience Canty Austin Hood](https://reader035.vdocuments.us/reader035/viewer/2022081603/56649d9d5503460f94a86b8c/html5/thumbnails/10.jpg)
2.
u = 2x + 1du = 2dxn = 15
Solution:
![Page 11: Integration Techniques Group Members Sam Taylor Patience Canty Austin Hood](https://reader035.vdocuments.us/reader035/viewer/2022081603/56649d9d5503460f94a86b8c/html5/thumbnails/11.jpg)
Trigonometric Integration and Natural Log Integration Formulas
![Page 12: Integration Techniques Group Members Sam Taylor Patience Canty Austin Hood](https://reader035.vdocuments.us/reader035/viewer/2022081603/56649d9d5503460f94a86b8c/html5/thumbnails/12.jpg)
Example of Trigonometric
Solution:
![Page 13: Integration Techniques Group Members Sam Taylor Patience Canty Austin Hood](https://reader035.vdocuments.us/reader035/viewer/2022081603/56649d9d5503460f94a86b8c/html5/thumbnails/13.jpg)
Example 2 of Trigonometric
OR
Solution: Solution:
![Page 14: Integration Techniques Group Members Sam Taylor Patience Canty Austin Hood](https://reader035.vdocuments.us/reader035/viewer/2022081603/56649d9d5503460f94a86b8c/html5/thumbnails/14.jpg)
Example of Natural Log u = 2x + 3du = 2dx n = -1
Solution:
![Page 15: Integration Techniques Group Members Sam Taylor Patience Canty Austin Hood](https://reader035.vdocuments.us/reader035/viewer/2022081603/56649d9d5503460f94a86b8c/html5/thumbnails/15.jpg)
Example 2 of Natural Log
Solution:
![Page 16: Integration Techniques Group Members Sam Taylor Patience Canty Austin Hood](https://reader035.vdocuments.us/reader035/viewer/2022081603/56649d9d5503460f94a86b8c/html5/thumbnails/16.jpg)
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:
![Page 17: Integration Techniques Group Members Sam Taylor Patience Canty Austin Hood](https://reader035.vdocuments.us/reader035/viewer/2022081603/56649d9d5503460f94a86b8c/html5/thumbnails/17.jpg)
Examples
1.
u =
Solution:
![Page 18: Integration Techniques Group Members Sam Taylor Patience Canty Austin Hood](https://reader035.vdocuments.us/reader035/viewer/2022081603/56649d9d5503460f94a86b8c/html5/thumbnails/18.jpg)
2.
Solution:
![Page 19: Integration Techniques Group Members Sam Taylor Patience Canty Austin Hood](https://reader035.vdocuments.us/reader035/viewer/2022081603/56649d9d5503460f94a86b8c/html5/thumbnails/19.jpg)
Integrating Powers of Sine and Cosine
Rules and Ways to integrate: Integrating Odd Powers
Integrating Odd and Even Powers Integrating Even Powers
![Page 20: Integration Techniques Group Members Sam Taylor Patience Canty Austin Hood](https://reader035.vdocuments.us/reader035/viewer/2022081603/56649d9d5503460f94a86b8c/html5/thumbnails/20.jpg)
Integrating Odd Powers
Solution:
![Page 21: Integration Techniques Group Members Sam Taylor Patience Canty Austin Hood](https://reader035.vdocuments.us/reader035/viewer/2022081603/56649d9d5503460f94a86b8c/html5/thumbnails/21.jpg)
Integrating Odd and Even Powers
Solution:
![Page 22: Integration Techniques Group Members Sam Taylor Patience Canty Austin Hood](https://reader035.vdocuments.us/reader035/viewer/2022081603/56649d9d5503460f94a86b8c/html5/thumbnails/22.jpg)
Integrating Even Powers
Half – Angle Formulas
Solution:
![Page 23: Integration Techniques Group Members Sam Taylor Patience Canty Austin Hood](https://reader035.vdocuments.us/reader035/viewer/2022081603/56649d9d5503460f94a86b8c/html5/thumbnails/23.jpg)
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)
![Page 24: Integration Techniques Group Members Sam Taylor Patience Canty Austin Hood](https://reader035.vdocuments.us/reader035/viewer/2022081603/56649d9d5503460f94a86b8c/html5/thumbnails/24.jpg)
Example
Solution:
![Page 25: Integration Techniques Group Members Sam Taylor Patience Canty Austin Hood](https://reader035.vdocuments.us/reader035/viewer/2022081603/56649d9d5503460f94a86b8c/html5/thumbnails/25.jpg)
Important/Unusual Integrals
+C
+C
+C
+C
![Page 26: Integration Techniques Group Members Sam Taylor Patience Canty Austin Hood](https://reader035.vdocuments.us/reader035/viewer/2022081603/56649d9d5503460f94a86b8c/html5/thumbnails/26.jpg)
Example 2
Solution:
![Page 27: Integration Techniques Group Members Sam Taylor Patience Canty Austin Hood](https://reader035.vdocuments.us/reader035/viewer/2022081603/56649d9d5503460f94a86b8c/html5/thumbnails/27.jpg)
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
![Page 28: Integration Techniques Group Members Sam Taylor Patience Canty Austin Hood](https://reader035.vdocuments.us/reader035/viewer/2022081603/56649d9d5503460f94a86b8c/html5/thumbnails/28.jpg)
Solution:
![Page 29: Integration Techniques Group Members Sam Taylor Patience Canty Austin Hood](https://reader035.vdocuments.us/reader035/viewer/2022081603/56649d9d5503460f94a86b8c/html5/thumbnails/29.jpg)
Example 2
Solution:
![Page 30: Integration Techniques Group Members Sam Taylor Patience Canty Austin Hood](https://reader035.vdocuments.us/reader035/viewer/2022081603/56649d9d5503460f94a86b8c/html5/thumbnails/30.jpg)
Definite IntegrationUsed when the
numerical bounds of the object are known
First Fundamental Theorem of Calculushttp://www.sosmath.com/calculus/integ/integ01/integ01.html
![Page 31: Integration Techniques Group Members Sam Taylor Patience Canty Austin Hood](https://reader035.vdocuments.us/reader035/viewer/2022081603/56649d9d5503460f94a86b8c/html5/thumbnails/31.jpg)
Examples of Definite Problems
Plug 2 inPlug 0 in
Solution:
![Page 32: Integration Techniques Group Members Sam Taylor Patience Canty Austin Hood](https://reader035.vdocuments.us/reader035/viewer/2022081603/56649d9d5503460f94a86b8c/html5/thumbnails/32.jpg)
Solution:
![Page 33: Integration Techniques Group Members Sam Taylor Patience Canty Austin Hood](https://reader035.vdocuments.us/reader035/viewer/2022081603/56649d9d5503460f94a86b8c/html5/thumbnails/33.jpg)
© 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