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Date Created: 05/27/15
Fall 2011 MA 16200 Study Guide Exam 1 Distance formula D 12 x12 I y2 yl2 22 zl2 equation of a sphere With cen ter h k l and radius 7 a h2 y k2 z 2 r2 gt Vectors in R2 and R3 displacement vectors PQ vector arithmetic components Standard basis vectors 1 hence 5 a1 a2 a3 ali l 0123 L312 length magnitude of a vector 2 Va a3 a3 dot or inner product of 5 and 3 51 albl a2b2 a3b3 properties of dot products Useful Vector v 7 cos 6 1 7 sin r 39 g a e a Angle between vectors cos6 a b lal lbl 01 G2 G3 Perpendicular orthogonal vectors direction cos1nes cosoz cos 6 cosy H a a a direction angles 05 6 y a a B a Vector projection of b onto 5 projab lt gt projection of b onto a at Work done by constant force is W F I3 1 i k Cross product 5 X I a1 32 a3 de ned w for vectors in R3 5 X J 5 and b1 b2 193 X J 6 other properties of cross products gtlt l 2 area of parallelogram spanned by 5 and b X as X l 2 area of triangle spanned by 5 and gt gt Cl 0 V 2 li gtlt 6 volume of parallelopiped spanned by 5 b 6 7 APPLICATIONS OF INTEGRATION b d a Areas Between Curves A f d3 or A f dy Y y X kY x my y d g c g x a b Note If curves cross7 you need to break up into several integrals b d b Volumes of Solids by Cross sectional Areas V Aa d513 or V Ay dy Where a Aa 2 area of the cross section of the solid With a plane J az aXis at the point 3 or Ay area of the cross section of the solid With a plane J y aXis at the point y y crosssectional area at x cross sectional area at y A00 d quotquot quot quot AW 0 Volumes of Solids of Revolution by DISK METHOD OR WASHER METHOD Use Disk Method or Washer Method When slices of area are perpendicular to axis of rotation In either case the cross section is always a diskwasher y xg y yf x Disk DiSk dx dY V f 7r Raciius2 dx 0r dy y Yfx ll ygx gtx b dx dy V f 7r Outer radius2 7r Inner radius2 dx 0r dy IMPORTANT When to use d3 or dy in Disk Washer method i If cross sections are J 1 axis7 use ii If cross sections are J y axis7 use d Volumes of Solids of Revolution by CYLINDRICAL SHELLS METHOD Use Cyclindri cal Shells Method When slices of area are parallel to axis of rotation Shell thickness is always either d3 or dy Y Y shell thickness dy w shell thickness dx thickness radius height b V 2 27r shell radius shell height shell thickness 0 e If force F is constant and distance object moved along a line is d then Work is W F d Here are the English and Metric systems compared Quantity English System Metric System Mass m slug 2 lb sec2ft kilogram kg Force F pounds lbs Newtons N 2 kg msec2 Distance d feet meters m Work W ft lbs Joules J 2 kg m2sec2 g 32 ftsec2 98 msec2 b If the force is variable say f1 then Work W f daj Hooke s Law f5a 2 km a work done compressing stretching springs emptying tanks pulling up chains 1 b baa f93df13 f Average of a function over an interval fave 1 Mean Value Thm for Integrals fave b a bf3 d3 fc for some a S c S b 8 TECHNIQUES OF INTEGRATION b b a Simple Substitution fg1g a dx 9 fudu let u g1 a 9a fud39v U vdu the LIATE rule b Integration by Parts let u 2 Log Invtrig Alg Trig Exp c Trig Integrals Integrals of the type sinm 1 cosquot xda and tanm 1 secquot 1 d3 Some useful trig identities i sin2 6 cos2 6 1 1 26 1 26 ii sin2 6 and cos2 6 C iii sin 26 2 sin 6 cos 6 iv tan2 6 1 2 sec2 6 Some useful trig integrals i ftanu du 1nsecu C ii fsecu du 1nsecutanu C d Trig integrals of the form sin ma sin 7151 d513 cos ma cos 7151 d513 sin ma cos 7151 d513 use these trig identities 1 sinA sinB E cosA B cosA B 1 cosA cosB E cosA B cosA B 1 sinA cosB E sinA B sinA B e Trigonometric Substitutions Empressi0n Trig Substituti0n Identity needed a2 332 masin6 1 sin26cos26 a2 12 matan6 1tan26sec26 5132 0 masec6 sec26 1ztan26 Or powers of these expressions
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