Theorem (Change of Variables Theorem):

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Agnes Bishop
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1 Avance Higher Notes (Unit ) Prereqisites: Integrating (a + b) n, sin (a + b) an cos (a + b); erivatives of tan, sec, cosec, cot, e an ln ; sm/ifference rles; areas ner an between crves. Maths Applications: Calclating areas an volmes of stanar geometric figres. Real-Worl Applications: Particle motion. Stanar Integrals From the new erivatives obtaine in the previos topic, we obtain a hanfl of new inefinite integrals for free. e e + C ln + C sec tan + C sec tan sec + C cosec cot cosec + C cosec cot + C In the secon integral, ( ) an ( < ). In other wors, is alwas or positive. M Patel (April ) St. Machar Acaem

2 Avance Higher Notes (Unit ) Differentials For a fnction f (), is the rate of change of with respect to. The qantities an in this epression o not have meanings on their own. However, confsingl, we can introce qantities like these which o have an inepenent meaning. A small incremental change in, enote b, will give rise to a small incremental change in, enote. The erivative is the limit of / as. We are intereste in a slightl ifferent qantit, namel, the change in the vales along the tangent line. f () Definition: For a fnction f (), the ifferential of f is the fnction efine b, ef f () Consiering the fnction, the efinition gives. This leas to an alternative wa of writing the ifferential of f. Another wa jst involves sing. M Patel (April ) St. Machar Acaem

3 Avance Higher Notes (Unit ) Corollar: The ifferential of f can be written in other was, f () It is the secon epression in the corollar that leas to a common misconception that the s cancel ot ; the on t cancel ot, an technicall is not to be treate as a fraction (becase it ain t). However, in practice, we can get awa with treating it as thogh it were a fraction. Eample Fin for the fnction. Eample Fin for the fnction +. + ( + ) Integration b Sbstittion It is not alwas obvios how to integrate a fnction. Sometimes, however, the fnction can be rewritten throgh a change of variable (i.e., a M Patel (April ) St. Machar Acaem

4 Avance Higher Notes (Unit ) sbstittion) in a form that is a stanar integral. The process involve in getting the stanar integral is calle integration b sbstittion. Theorem (Change of Variables Theorem): b f (( ()) () a f () c where c (a) an (b). is the sbstittion variable. To ecipher this will probabl reqire some eamples. The general iea is to spot something in the integran that is the erivative of something else (the sbstittion variable ) in the integran. Obvios Sbstittions Eample Fin e +. Note that ifferentiating + gives, which is, apart from a factor of, eqal to. So, let () will give f ( ()) Theorem, +. Then, (). f () e e +. So, accoring to the Change of Variables e + e e + e e + C e + + C Normall, we on t eplicitl se the etestable formla given in the theorem. In practice, we work with the ifferentials. So, in the eample M Patel (April ) St. Machar Acaem

5 Avance Higher Notes (Unit ) above, we wol jst write, solve for an then go straight to the secon line in the above calclation. Eample Fin + + ( > ). Let +. Then ( + ). The new limits are () an (). So, + + ln ln () ln () ln ln Sbstittion eamples involving efinite integrals can be one in was. Like Eample, the new limits are worke ot an the integration is one with the new variable an the limits are sbstitte in. Alternativel, the integration with the new variable is one an the answer written in terms of the ol variable (jst like inefinite integrals) an the ol limits are sbstitte in. Which approach is better epens on the integral an personal preference. + + () () ln () () M Patel (April ) St. Machar Acaem

6 Avance Higher Notes (Unit ) ln ( + ) ln () ln () ln ln There are lots of ifferent tpes of sbstittion qestions. Eamples an are specific cases of general tpes of eas sbstittion. ( Df ) f f + C Df ln f + C f Remember that Df means f. The ifficlt in sbstittion qestions is eciing which fnction to replace b. Choose the one that ifferentiates to give something else (apart from possibl a nmerical factor) in the integran. In the following eamples, tr choosing to be the other fnction to see what goes wrong. Eample Evalate π cos sin e Note that D (cos ) sin. So, let cos, Then sin, so sin. Also, () an (π). So, π cos sin e e M Patel (April ) 6 St. Machar Acaem

7 Avance Higher Notes (Unit ) e e ( e ) ( e ) e e Eample 6 Integrate ( 9). Let 9. Then 8. Ths, ( 9) C 8 ( 9) + C Eample 7 Integrate 7 7 cosec. Let 7. Then 7 7. Hence, 7 7 cosec 7 cosec 7 ( cot ) + C 7 cot 7 + C M Patel (April ) 7 St. Machar Acaem

8 Avance Higher Notes (Unit ) Eample 8 π Evalate cos sin. Let sin. Then cos. In this efinite integral, we will write the answer in terms of an se the limits (tr it the other wa). We have, π cos sin π sin Eample 9 Fin cos. Let. Then, so that. So, cos cos (sin ) + C sin + C M Patel (April ) 8 St. Machar Acaem

9 Avance Higher Notes (Unit ) Difficlt Sbstittions - Rewriting the Integran More ifficlt qestions involve rewriting the integran before or after the sbstittion phase. Normall, the sbstittion is given. Eample Integrate cot sing the sbstittion sin. Recalling the efinition of cot, cot cos sin Given sin, D (sin ) cos. Then cos. So, cot ln + C ln sin + C Note the alternative wa of writing Eample in the first line. Integrate Note that ( sin ) cos sing the sbstittion sin. cos cos. cos cos cos ( sin sin ) C for cos an S for sin, the integral becomes, (cos ) cos +. Using the abbreviations cos C ( S S + ) Let sin. Then cos. Hence, M Patel (April ) 9 St. Machar Acaem

10 Avance Higher Notes (Unit ) cos ( + ) + + C sin sin sin + + C Eample Evalate sing the sbstittion ( ). Also, () an (). Ths, + ( ) ( + ) ln 8 ln Fill in the etails for the last eqalit. M Patel (April ) St. Machar Acaem

11 Avance Higher Notes (Unit ) Areas Recall that the area bone b a crve, the lines a an b an the ais is given b, A b f () a where the crve lies above the ais an a < b. Area between Crve an the ais The area bone b a crve, an the ais can also be calclate. f () c A a b Theorem: The area bone b the crve f (), the lines c an ( c < ) an the ais where the crve lies to the right of the ais is given b, A f () c Eample Fin the area bone b the crve an., the ais an the lines M Patel (April ) St. Machar Acaem

12 Avance Higher Notes (Unit ) Since the vales are both positive, the inverse fnction we are intereste in is. Hence, A.... Area between Crves an the ais The area between crves an the ais can be fon b a similar procere to that between the ais. Jst remember to invert the fnctions an integrate right-han fnction mins left-han fnction. Theorem: The area between crves f () an g () (f () g () ), the ais an the lines c an ( c < ) is given b, A (f () g ( ) c f () c g () c Eample Fin the area of the region bone b the crves the ais. an an M Patel (April ) St. Machar Acaem

13 Avance Higher Notes (Unit ) The fnctions meet when an, or, in terms of vales, an. The inverte fnctions are an. A qick sketch shows that is the right-han fnction. So, A.. Check that the same answer is obtaine b integrating between the ais. Volmes of Soli of Revoltion Integration is also se to calclate volmes of regions. Definition: A soli of revoltion is a D shape forme b rotating a crve 6 abot either the ais or ais. Imagine part of the graph of a fnction rotate aron either the ais or ais. The reslting figre is the shape whose volme we want to fin. This volme is calle the volme of soli of revoltion. This techniqe can be se to erive formlae sch as the volme of a cone, sphere an lots of other fnk D shapes. There are cases to consier. M Patel (April ) St. Machar Acaem

14 Avance Higher Notes (Unit ) Volme Generate b a Crve abot the -ais f () b a The iagram shows the graph of f () rotate 6 abot the ais between a an b. The volme reqire is the region between the otte vertical lines an the circles (which look like ellipses above e to perspective). Theorem: The volme of soli of revoltion generate b rotating the crve f () 6 abot the ais between a an b (a < b) is given b, V π b a Eample Fin the volme of soli of revoltion between the lines an when is rotate 6 abot the ais.. So, V π π M Patel (April ) St. Machar Acaem

15 Avance Higher Notes (Unit ) π ( ) π ( ) π Volme Generate b Crves abot the -ais The volme of a region forme between crves b rotating them abot the ais can be fon b a similar techniqe to that in the previos theorem. Theorem: The volme of soli of revoltion forme between crves an ( < ) b rotating them 6 abot the ais is given b, b V π ( ) a where the crves intersect at a an b (a < b). Eample 6 Fin the volme of soli of revoltion obtaine b rotating the crves an 6 6 abot the ais. The crves meet when 6 ( 6) an. The top crve is 6. So, V π ( 6 ) M Patel (April ) St. Machar Acaem

16 Avance Higher Notes (Unit ) π π.6 π π ( 6 ) 6 π Volmes abot the ais are calclate in a similar manner to that abot the ais. Volme Generate b a Crve abot the -ais Theorem: The volme of soli of revoltion generate b rotating the crve f () 6 abot the ais between c an (c < ) is given b, V π c Eample 7 Calclate the volme of soli of revoltion generate between an 9 b rotating the crve 6 abot the ais. V π 9 M Patel (April ) 6 St. Machar Acaem

17 Avance Higher Notes (Unit ) π 9 π Volme Generate b Crves abot the -ais Theorem: The volme of soli of revoltion forme between crves an ( < ) b rotating them 6 abot the ais is given b, V π ( ) c where the crves intersect at c an (c < ). Eample 8 Calclate the volme of soli of revoltion obtaine b rotating the crves an 6 6 abot the ais. From Eample 6, the crves meet at an, eqivalentl, an 6. The right-han crve is. Hence, V π 6 96 π π π 8 6 M Patel (April ) 7 St. Machar Acaem

18 Avance Higher Notes (Unit ) π (6 6) 8 π Rectilinear Motion The formlae for acceleration an velocit lea to formlae for velocit an isplacement respectivel (b thinking of integration as the opposite of ifferentiation). v (t) s (t) a (t) t v (t) t The whole stor can b smmarize b the following iagram. s (t) v (t) a (t) Definition: A particle is at rest when it has velocit. A particle is initiall at rest when it has velocit at time. Eample 9 A particle initiall at rest has acceleration escribe b the formla a (t) t + t. If it initiall has zero isplacement, calclate the isplacement, velocit an acceleration after secons. a () () + () a () 9 m/s. The velocit is, t v (t) ( t + t ) v (t) t + t t + C M Patel (April ) 8 St. Machar Acaem

19 Avance Higher Notes (Unit ) The constant of integration is crcial here. When t, v. Hence, C. So, v (t) t + t t v () () + () () v () m/s. The isplacement is, t s (t) ( t + t t ) s (t) t + t t + D When t, s, Hence, D. Ths, s (t) t + t t s () () + () () s () 8 m. M Patel (April ) 9 St. Machar Acaem

Logarithmic, Exponential and Other Transcendental Functions

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