ENGINEERING MATH II
ENGINEERING MATH II MATH 152
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This 4 page Class Notes was uploaded by Vivien Bradtke V on Wednesday October 21, 2015. The Class Notes belongs to MATH 152 at Texas A&M University taught by Y. Gorb in Fall. Since its upload, it has received 45 views. For similar materials see /class/226007/math-152-texas-a-m-university in Mathematics (M) at Texas A&M University.
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Date Created: 10/21/15
MATH 152 Fall 2008 Yuliya Gorb goermath tamu edu wwwmathtamu edu gorbmath152ia112008html Formula Sheet for Exam 1 Area The area of the region bounded by y and y 91 and the lines 1 a 1 b for continuous functions7 where 2 for all 1 in a z is b Avmemmm Volume H The volume of the solid that lies between planes 1 a 1 b and whose crosssection perpendicular to the 1axis has an area A1 is V ab A1d1 to 1 Disk method The volume of the solid of revolution that lies between planes 1 a 1 b and which is generated by revolving of y around the 17axis is V ab 7rf12d1 i Washer method The volume of the solid of revolution that lies be tween planes 1 a 1 b and which is generated by by revolving of the region between y and y 91 2 around the 17axis is CA3 b v wvwemm m 4 Method of cylindrical shells The volume of the solid generated by the revolving of the region bounded by y 1 a and 1 12 around yiaxis is b V 27TIfIdI Here 1 is the radius of the shell7 is its height Work The work done in moving an object from a to b by the force is W Abf1d1 1 Hook s law f I 7m where k is the spring constant7 z is the displacement from equilibrium position A erage The average of the function on the interval a b is 1 b fave a Mean Value Theorem lf is continous on the interval a b then there exists a number 5 in m 12 such that b mm fcb 7 a Integration by Parts fIy IdI f191f I9IdI udvuv7vdu b b b agmm fltzgtgltzgtla7 f ltzgtgltzgtdz For de nite integrals Trigonometric Integrals ll sinm 1 cos zdz o nis odd n2kl a save one cosine factor b use cos2 I l 7 sin2 I c make substitution u sin I du cos zdz o mis odd m2kl a save one sine factor b use sin I l 7 cos2 I c make substitution u cos I du 7 sin zdz 0 both m and n are even m 216 n 2k use 2 cos2 I lcos 21 sin I l7cos 21 sinz cosz sin 21 2 tanm 1 sec zdz o n is even n 216 a save a factor of sec2 1 b use sec2 1 1tan2 z c make substitution u tan 1 du sec2 zdz o mis odd m2k1 a save a factor of sec 1 tanz b use tan2 1 sec 1 71 c make substitution u sec 1 du sec 1 tan zdz Important Trigonometric Identities sinz 1 tanz secz cscz cos 1 cos 1 sin I sin2ercos2117 tan211sec21 2 cos I 1 cos 217 sin2 I 17 cosZI7 sinZI 2sinzcosz d 2 d 7tanz sec 1 7secz secztanz dz dz tanzdzlnlseczl07 seczdzln secztanzlC 1 sinz cosy sin1 y Sinz 1 cosz cosy cos1 y cosz sinz siny cosz 7 y 7 cosz Substitution M Qz Partial Fraction Decomposition for rational function fx Where Pz and are polynomials 1 1f deg Pz 2 deg then use long division of two polynomials to get P I R I Where Rz is a remainder such that deg Rz lt deg and 51 is a quotient 2 CA3 Express the proper rational function Factor the denominator as a product of linear factors of the form az b and irreducible quadratic factors of the form 0412 121 c Where b2 7 4ac lt 0 31 QW as a sum of partial fractions of the form AIB or a12 121 cm L ax bk a if is a product of distinct linear factors ie mm b1agz 122 l l akz bk then there exist constants A1 A2 Ak such that Rm QW A1 alz 121 7a2xb2m Ak akz bk if is a product of linear factors some of Which are repeated elgl has a factor ax by the corresponding partial fractions are A1 A2 A2 azbazb2 uazbi if contains irreducible quadratic factors none of Which is re peated ilel a112 1211 cl 212 1221 62Hiam12 bmz cm then there exist constants A1 Am and B1 H l Bm such that 131 7 A11 B1 A21 B2 Amz Bm 7 a112blzcl l a212bgz02 l l am12bmzcm if contains a repeated irreducible quadratic factor elgl has a factor ale 1211 01V Where b2 7 4ac lt 0 then the corresponding partial fractions are A11 B1 A21 B2 Air BI a12 121 5 a12 121 c2 Hi a12 121 c
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