B-9= B.Bà B 68 B.Bß B/.Bß B=38 B.B >+8 ab b.bß ' 68aB b.bà

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Suzan Bridges
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1 8.1 Integration y Parts.@ a.? a Consider. a? a œ? a Þ....@ a..? a We can write this as? a œ a? a@ a Þ... If we integrate oth sides, we otain.@ a a.?. œ a? a..? a. or...? a.@ œ? a.?þ If + and, are in the interval where? a a are continuos, and the integrals are defined, from + to,,,,? a.@ œ? al + + a.?þ eg 1 a a ˆ 9=.à 68.ß /.ß =38. >+8 a.ß 68a.à a 68a.à / =38a. eg 2 Find the volume of revolution aout the axis of the area in the first quadrant of C œ 68a and C œ 1. eg 3 Find the volume of revolution aout the axis of the area in the first quadrant of C œ 68a and œ 1. Practice 1) /. œ / Ð ÑG ( Ñ 68 a. œ Ð68ÐÑ ÑG ( ( 3) =38 ÐÑ. œ =38 ÐÑ G 4) =38ÐÑ œ 9=ÐÑÐ=38ÐÑ9=ÐÑÑG 5) 68ÐÑ 68ÐÑ. œ G 6) / 9= a. œ / Ð=38ÐÑ9=ÐÑÑG 8.2 Trigonometric Integrals A. Integrals of the form =38 a 9= a 9= a 9= a +Þ If 7 and 8 are even, use the half angle formulas =38 a œ ß 9= a œ Þ eg 4 Find the volume generated when C œ 9= a is rotated aout the axis in [0, 1/2]. 1 Î 1 1 Î y the Disks œ % 1 9=. œ 1 =38 a a 9= a. œ 21Š l = 1?Þ?83>=,Þ If 7 is odd and8 even, save a sine and make the rest of the factors cosines y using a Pythagorean identity. & eg 5 =38 a9= a. œ ˆ 9= a 9= a =38a. œ 9= a 9= a & Þ If 7 is even and8 odd, save a cosine and make the rest of the factors sine y using a Pythagorean identity.

2 ˆ =38 a &Î 9= a eg 6 a =38 a. œ a =38 9=. œ =38 œ a =38a & =38a ˆ &=38 a & 7 8. Integrals of the form sec a tan a +Þ If 7 is even, save a =/ a and make the rest tangent y using a Pythagorean identity. ) eg 7 >+8 & a =/ % a. œ >+8 & a ˆ >+8 a =/ a. œ >+8 a >+8 a ) eg 8 =/ % a. œ ˆ >+8 a =/ a. œ >+8 a >+8a,Þ If 8 is odd, save a =/a >+8a and make the rest secant y using a Pythagorean identity. eg 9 a a ˆ a & >+8 a >+8 a >+8 =/. œ =/ =/ a =/a >+8a. œ ) ). If 7 œ and 8 even, save a >+8 a and change the rest to secant y using a Pythagorean identity. eg 10 a ˆ >+8. œ =/ a. œ >+8a eg 11 Find the area ounded y C œ >+8 % a and theaxis etween œ and œ 1Î%Þ 1 Î% % 1 a Î% ˆ a 1 a Î% >+8. œ =/ >+8. œ =/ a >+8 a>+8 a. œ 1Î% a a ˆ >+8 1Î% a 1 =/ >+8 =/. œ Š a >+8a l œ.. If 8 œ and 7 odd, do integration y parts. % eg 12 =/ a. œ =/a >+8a =/a >+8 a. œ =/a >+8a =/a ˆ =/ a. œ =/a >+8a =/ a. =/a.þ Since =/ a. œ =/a >+8a =/ a.68l=/a >+8alß y solving for a =/. we otain =/ a. œ a=/a >+8a68l=/a >+8al 7 8 C. Integrals of the form csc a cot a +Þ If 7 is even, save a csc a and make the rest cotangent y using a Pythagorean identity. ( & eg 13 9> % a= % a. œ 9> % aˆ 9> a = a. œ 9> a 9> a ( &,Þ If 8 is odd, save a =a9> a and make the rest cosecant y using a Pythagorean identity. eg 14 9> a= a. œ ˆ = a = & a=a9> a. œ = ) 9> a a )

3 Practice: Ñ % >+8 >+8 a. œ >+8 Ñ >+8 a =/ =/ & =/ a. œ Ñ >+8 a =/ & % tan tan a. œ & %Ñ >+8 a =/ a. œ =/>+8 68l=/ >+8l &Ñ 9> a = = & = a. œ & Ñ = a. œ 68l= 9>l (Ñ 9> % a = 9> = % a. œ % )Ñ 9> 9> a. œ 68l=38l 8.3 Trigonometric Sustitutions A) Integrals odd powers of +, use œ +=38a) as sustitution. eg 15 Find. If we use the sustitution œ =38a) we find 9= a) a a ˆ a. œ =38 9= ).) œ 9> ).) œ = ).) œ a) 9> a) ) œ =38 a after making a triangle with angle ). ) Integrals odd powers of +, use œ +>+8a) as sustitution. eg 16 Find. If we use the sustitution œ tan a) we find. =/ a) 9= a) œ.) œ.) œ =a) œ ß after making a triangle with angle ). >+8 a) =/ a) =38 a) C) Integrals odd powers of +, use œ +=/a) as sustitution. eg 17 Find. % If we use the sustitution œ =/a) we find. =/ a) >+8a) % œ.) œ =/ a). ) œ 68l=/a) >+8a) l œ 68l lß after % >+8 a) making a triangle with angle ). eg 18 Find *. If we use the sustitution œ =/a) we otain * 1 Î%. œ >+8 ).) œ Ð11Î%Ñ

4 eg 19 Find 0 Î. a* Î If we do the sutitution œ >+8) ß we otain Î. 1 Î% Î œ 9=).) œ Þ 0 a* 0 eg 20 Find the volume of revolution aout the axis of the area in the first quadrant of C œ 1 1 and the line œ Þ Ans: Ð Ñ % Practice 1). % = 4 %. * œ * * % % %. Î &.. ÐÑ * 1Î 9= 0. œ 68l l =38. œ a% Î % %. =68Ð Ñ %& Ñ ˆ Ñ. œ 68l lg %Ñ œ 68l l68l l &Ñ œ œ +<>+8ˆ 6Ñ (Ñ 0 )Ñ 8.4 Integration y Partial Fractions Case I Distinct Linear Factors ÐL/+@3=3./ G9@/<?: Q/>29.Ñ eg 20 Find. a a. y partial fractions E F G a a a a œ Þ a a a a œ. a a a a y the coverup method E œ l œ à F œ l œ à G œ l œ So. œ. œ 68l l. a a a a eg 21 Find a a a a. Î Î œ œ a a a a a a a a Î Î a a a a Since y long division, and y the Coverup method,. œ. œ 68 Î or

5 Case II Repeated Factors eg 22: Find.. a E F G H a a a a y partial fractions = where ˆ ˆ E œ l œ à F œ l œ à G œ l œ à H œ l œ a x x so =, or. œ a a a a a. œ 68l l a a a a a eg 23 Find %. E F G H y partial fractions œ = where % a a a w a E œ l œ à G œ l œ à F œ l œ &à H œ l œ & a x x or = & &, so. œ a a a a & & & a a a w Š Š. œ 68l l w ww eg 24 Find %. This integral can e found using partial fractions, ut if we look closely we can see that if %? œ ß.? œ % ˆ.? %.ß so. œ œ 68l l Case III Irreducile Quadratic Factors eg 25 Find.Þ a % %? % Since E FG a œ a, y coverup on the first term we otain E œ l œ Þ 3 a y coverup on the second term and evaluated at either root we otain ± œ FGl Ê 3 œ Fa3G Ê 3 œ G F3 3 3 If we equate real and imaginary parts, we otaing œ ßF œ Þ Since the partial fraction ll expansion of œ,. œ. œ 68 >+8 a. eg 26 Find a a a a a. y using systems of equations. E FG 1 FG a a a a Since œ, y coverup E œ so œ. If we multiply y the LCD of all the fractions, we otain œ afg or

6 œ afg. If we evaluate the expression at any numer, say œ, we otain F œ GÞ If we evaluate the same expression at another numer, say œ, we otain F G œ Þ If we solve the system, we have G œ andf œ Þ Since the partial fraction expansion of ll œ,. œ. œ 68 >+8 a. a a a a a a *.Þ EF GH a a œ * a a * eg 27. Find Since, y coverup on the first term we otain a * 3 3 If we equate real and imaginary parts, we otain 8 8 and y coverup on the first term evaluating at either root 3, we otain ± œ FGl Ê 3 œ F3 G Þ ß F œ ß G œ Þ y coverup on the second term evaluating at either root 33, we otain ± œ FGl Ê 3 œ F3 G Þ If we equate real and imaginary parts, we a otainß F œ ß and G œ Þ 8 Î) Î) a a * a a * a a. œ * ) a a. œ 68 * ) * Since the partial fraction expansion of œ, Š asic Integration Procedures 1) asic Sustitutions +Þ. œ ß y the sustitution? œ.,þ. œ 68l lß? œ y the sustitution. Þ. œ >+8 a a Try. ANS: >+8 a/ / / 2) Completing the Square +Þ. œ. œ. œ =38 ˆ % ) ) Éa% a.>.> >> É a>.> > %>,Þ. œ. œ >+8 ˆ Þ œ œ =38 a> Try ANS: =38 a> 2 3) Trigonometric Identities %

7 +Þ >+8a=a. œ =/a. œ 68l=/a >+8al,Þ = =38. œ a a 9> a. œ 68l=38al Þ a=/a>+8a. œ ˆ =/ a=/a >+8a>+8 a. œ ˆ =/ a=/a >+8aÒ=/ aó. œ >+8a=/a Trya=/a9> a. ANS: >+8a œ 68l=a 9> al9> a 4) Reducing Improper Fractions +Þ (. œ ˆ. œ 68ll,Þ. œ >+8 a Þ. œ. œ 68l l % Try ˆ. ANS: >+8 % 5) Separating Fractions +Þ. œ Š. œ =38 a 1Î%=38a 9= a 1/2 ) 1 0 ANS: a %. 68 %,Þ. œ Š. œ 68a 1Î% Þ. œ >+8a=/al œ Try 6Ñ Sustitutions An integral containing a term :Î; a +,, let? œ a +, +Þ. Î p? œ a ß? œ ß?.? œ. let so. 2? œ.? œ 68l l œ 68l l??. Î% %.? Î Î% let so Î Î%?? Î Î% Î% Š?,Þ p? œ ß? œ ß %?.? œ.à œ %.? œ %?.? œ %?68l?l œ % %68l l Þ 1 Î. œ let? œ a ß? œ ß?.? œ. so 1. a a 2? a??????? Î; œ.? œ.? œ 68l l œ 68l l Try a 8.8 Improper Integrals. ANS: 68l l

8 The definition of the Riemann integral, + 0a. as a finite limit of a Riemann sum can only e used if the interval of integration is finite and the function is ounded on that interval. Improper integrals have unounded functions in interval of integration or infinite intervals of integration. Improper integral can Converge, Diverge to _ or Diverge without limit. eg 28 a) Improper integrals of unounded functions 5. 2 *.. 68a. eg 29 ) Improper integrals with infinite intervals of integration _ : : diverges à. converges to 1 à. converges to for : ā à =38 a.. à diverges without limit; converges to 1. diverges to _ ; / /.. œ diverges without a limit;. œ diverges to _; converges to /2. 1 Find the area ounded y 0a œ in [3,_]; area ounded y 0a œ in [0,1] Þ Comparison Test for Improper Integrals. Let 0a and 1a e continuos functions such that 0a ā 1a for +Þ If 0a. converges, 1a. converges. If 1a. diverges, 0a. diverges. eg 30 _ / / _ 1. diverges since, ā and. diverges. (: œ Ñ 1 eg 31 converges since, and.. converges. (: ā Ñ 0 0 eg 32 / / converges since, and.. converges.

4-3 Trigonometric Functions on the Unit Circle

4-3 Trigonometric Functions on the Unit Circle Find the exact value of each trigonometric function, if defined. If not defined, write undefined. 9. sin The terminal side of in standard position lies on the positive y-axis. Choose a point P(0, 1) on