Calculus Cheat Sheet - Pauls Online Math Notes

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Calculus Cheat Sheet - Pauls Online Math Notes

Definite Integral: Suppose f ( )on [ ab , ] . Divide [ , ]widthD x and chooseCalculus Cheat SheetIntegralsDefinitionsx is continuousab into n subintervals ofb*Then ( ) limå( i )a¥n®¥ i = 1*xifrom each interval.ò f x dx = f x D x .Anti-Derivative : An anti-derivative of f ( x )is a function, F( x ) , such that F¢ ( x) = f ( x).Indefinite Integral : ò f ( x) dx= F( x)+ cwhere F( x ) is an anti-derivative of f ( x ) .Fundamental Theorem of Calculusf x is continuous on [ ab , ] then Variants of Part I :xd ux ( )g( x) = ò f () t dt is also continuous on [ ab , ]f () t dt u ( x) f u( x)adxò = ¢ éaë ùûd xd bg¢ x = f t dt = f xdxò .f () t dt v ( x) f v( x)adxò =- ¢ évx ( ) ë ùûf x is continuous on[ ab , ] , F( x ) isd ux ( )f () t dt u x f ux v x fdxò = ¢ - ¢vx ( )F x = ò f x dx )Part I : If ( )and ( ) () ( )Part II : ( )an anti-derivative of f ( x ) (i.e. ( ) ( )bò f x dx F b F a .athen ( ) = ( ) - ( )( ) ± ( ) = ( ) ± ( )ò ò òò ± = ò ± òòf x g x dx f x dx g x dxb b b( ) ( ) ( ) ( )f x g x dx f x dx g x dxa a aa( ) 0a f x dx =ba( ) =- ( )ò òf x dx f x dxbabaIf f ( x) ³ g( x)on a£ x£ bthen ( ) ³ ( )ò òbIf f ( x ) ³ 0 on a£ x£ b then ( ) 0òaPropertiescf x dx=c f x dxb( ) ( )( ) [ ( )] ( ) [ vx ( )]ò ò , c is a constantbbò cf ( x) dx=cò f ( x)dx , c is a constantaba( ) = ()ò òòf x dx g x dxa f x dx ³baf x dx f t dtb( ) £ ( )baòf x dx f x dxbIf m£ f ( x)£ M on a£ x£ b then mb ( - a) £ f ( x) dx£ M ( b-a)òaaacos sinCommon Integralsò kdx= kx+cò udu = u+cu uò e du = 2+ cò csc udu =- cot u+cx n1 n 1, 1n 1+ò = ++cn ¹- ò sinudu =- cosu+c-1 1 2òx dx= ò x dx= ln x + c ò sec udu = tan u+c1ò dx= 1a ln ax+ b + cax+bò lnudu = uln ( u)- u+cò secutanudu = secu+cò cscucotudu =- cscu+cVisit http://tutorial.math.lamar.edu for a complete set of Calculus notes.òòòòtanudu = ln secu + csecudu = ln secu+ tan u + cu( )1 du 1 -1= tana2 u2 a a + c+1u( )du -1= sina2 u2a+ c-© 2005 Paul Dawkins


Calculus Cheat SheetStandard Integration TechniquesNote that at many schools all but the Substitution Rule tend to be taught in a Calculus II class.u Substitution : The substitution u = g( x)will convert ( )( )du = g¢x dx . For indefinite integrals drop the limits of integration.22 3Ex. ò ( )5x cos x dx1u = x Þ du = 3xdx Þ xdx=du3 2 2 13131 1 :: 232 8x= Þ u = = x= Þ u = =Integration by Parts : òudv= uv-ò vdu andòbgbò f ( g x ) g¢ ( x) dx=ò f ( u)du usinga( )( )ga( ) = ò ( )585= sin( u) = ( sin( 8) -sin()1)2 82 351 1 3integral and compute du by differentiating u and compute v using vEx.ò-xò x dx- x - xu = x dv= e Þ du = dx v=-eò- x - x - x - x - xxe dx=- xe + e dx=-xe - e + c5x cos x dx cos u du3 1 3bb bò udv = uv - vduaa ò . Choose u and dv fromaEx.ò53ln xdx= ò dv .u = ln x dv= dx Þ du = dx v=xò1x5 5 5 5( ) ( )( ( ) )lnxdx= xlnx - dx= xlnx -x3 3 33= 5ln 5 -3ln 3 -2Products and (some) Quotients of Trig Functionsn mn mFor ò sin xcosxdx we have the following : Forò tan xsecxdx we have the following :1. n odd. Strip 1 sine out and convert rest to2 2cosines using sin x= 1- cos x, then usethe substitution u = cos x.2. m odd. Strip 1 cosine out and convert rest2 2to sines using cos x= 1- sin x, then usethe substitution u = sin x.3. n and m both odd. Use either 1. or 2.4. n and m both even. Use double angleand/or half angle formulas to reduce theintegral into a form that can be integrated.1. n odd. Strip 1 tangent and 1 secant out andconvert the rest to secants using2 2tan x= sec x- 1, then use the substitutionu = sec x.2. m even. Strip 2 secants out and convert rest2 2to tangents using sec x= 1+ tan x, thenuse the substitution u = tan x.3. n odd and m even. Use either 1. or 2.4. n even and m odd. Each integral will bedealt with differently.sin 2x 2sin x cos x2cos x11 cos 2x2sin x1= 1-cos 2x( )Trig Formulas : ( ) = ( ) ( ) , ( ) = + ( ) , ( ) ( )2ò2( )Ex.òòtanxsec3 5xdx= ò2 4ò( sec x 1sec ) xtanxsecxdx2 4ò( u 1) udu ( u sec x)3 5 2 4tan sec tan sec tan secx xdx x x x xdx= -= - == sec x- sec x+c1 7 1 57 5Ex.òsin5xdx cos3x5 42 2sin x sin xsinx (sin x)sin x= =cos3x cos3x cos3x(1-cos 2x)2sinx= ò=cos3x(1-u2) 21 2u2 u4=- du- +ò =-u3 ò u3ò ò òdx dx dx( cos )dx u xdu= sec x+ 2ln cosx - cos x+c1 2 1 22 2Visit http://tutorial.math.lamar.edu for a complete set of Calculus notes.© 2005 Paul Dawkins


faCalculus Cheat SheetApplications of IntegralsNet Area : ò ( x ) dx represents the net area between f ( )x and thex-axis with area above x-axis positive and area below x-axis negative.Area Between Curves : The general formulas for the two main cases for each are,b( ) éupper functionù éëlower functionùò ëûû & = ( ) Þ = ëright functionû-ëleft functionòy = f x Þ A= - dxadé ù é ùûcx f y A dyIf the curves intersect then the area of each portion must be found individually. Here are somesketches of a couple possible situations and formulas for a couple of possible cases.bcbò ( ) ( ) A= ò f ( y) -g( y)dyc= ( )- ( ) + ( )-( )A= f x -g x dxaVolumes of Revolution : The two main formulas are V ( )dò òA f x g x dx g x f x dx=ò A x dx and V =ò A( y)dy . Here issome general information about each method of computing and some examples.RingsCylindersA = π (( outer radius) 2 - ( inner radius)2)A=2π( radius)( width / height)Limits: x/y of right/bot ring to x/y of left/top ring Limits : x/y of inner cyl. to x/y of outer cyl.f x ,f y ,f y ,f x ,Horz. Axis use ( )g( x ), A( x ) and dx.Vert. Axis use ( )g( y ), A( y ) and dy.Horz. Axis use ( )g( y ), A( y ) and dy.acVert. Axis use ( )g( x ), A( x ) and dx.Ex. Axis : y = a> 0 Ex. Axis : y = a£ 0 Ex. Axis : y = a > 0 Ex. Axis : y = a£0outer radius : a-f ( x)inner radius : a-g( x)outer radius: a + g( x)inner radius: a + f ( x)radius : a-ywidth : f ( y) - g( y)radius : a + ywidth : f ( y) - g( y)These are only a few cases for horizontal axis of rotation. If axis of rotation is the x-axis use they = a£ 0 case with a = 0. For vertical axis of rotation ( x= a> 0 and x= a£ 0) interchange x andy to get appropriate formulas.Visit http://tutorial.math.lamar.edu for a complete set of Calculus notes.© 2005 Paul Dawkins


Work : If a force of F( x)moves an objectin a x b£ £ , the work done is W ( )= òCalculus Cheat SheetbaF x dxAverage Function Value : The average value1 bof f ( x ) on a£ x£ b is favg= ò f ( x)dxb-aaArc Length Surface Area : Note that this is often a Calc II topic. The three basic formulas are,bL= ò ds SA= ò 2πyds (rotate about x-axis) SA= ò 2πxds (rotate about y-axis)ab awhere ds is dependent upon the form of the function being worked with as follows.2dy( dx ) ( )2dx( ) ( )ds = 1 + dx if y = f x , a£ x£bds = 1 + dy if x= f y , a£ y£bdyb a2dx 2 dy( ) ( ) () ()ds = + dt if x = f t , y = g t , a £ t £ bdtdtdr 2( ) ( )2ds = r + dθ if r = f θ , a£ θ £ bdWith surface area you may have to substitute in for the x or y depending on your choice of ds tomatch the differential in the ds. With parametric and polar you will always need to substitute.Improper IntegralAn improper integral is an integral with one or more infinite limits and/or discontinuous integrands.Integral is called convergent if the limit exists and has a finite value and divergent if the limitdoesn’t exist or has infinite value. This is typically a Calc II topic.Infinite Limit¥1. ( ) limtbbò f x dx=ò f ( x)dx2. ò f ( x) dx=lim ò f ( x)dxat®¥¥ ¥c3. ( ) = ( ) + ( )aa+t®atθ- ¥t®-¥ò f x dx f x dx-¥ ò-¥ò cf x dx provided BOTH integrals are convergent.Discontinuous Integrandbbbt1. Discont. at a: ò f ( x) dx=lim ò f ( x)dx 2. Discont. at b : ò f ( x) dx=lim f ( x)dx- ò3. Discontinuity at a c bb c b< < : ( ) = ( ) + ( )a a catt®bò f x dx ò f x dx ò f x dx provided both are convergent.Comparison Test for Improper Integrals : If f ( x) ³ g( x) ³ 0 on [ , )¥a ¥ then,1. If ò ( ) conv. then ò ( ) conv. 2. If ò ( ) divg. then ( )a f x dxUseful fact : If a > 0 thenbFor given integral f ( x ) dxadivide [ , ]¥a g x dx¥1ò ap dx converges if 1x¥a g x dxp > and diverges for p £ 1.¥aò f x dx divg.aApproximating Definite Integralsb-aò and a n (must be even for Simpson’s Rule) define D x =nandab into n subintervals [ x , x ] , [ x , x ] , … , [ x x ]0 11 2-with x0n 1 , nb* * *Midpoint Rule : ( ) »D é ( 1) + ( 2) + + ( n)= a and xn= b then,f x dx x f x f x f x *òùa ë⋯û, ixi-1 , xib Dxò f x dx» éf x0 + 2f x1 ++ 2f x2 + + 2f xn1f xna-+ ù2ë⋯ûb Dxò f x dx» éf x0 + 4f x1 + 2f x2 + + 2f x24f xa-+-1+ f x3ë⋯x is midpoint [ ]Trapezoid Rule : ( ) ( ) ( ) ( ) ( ) ( )Simpson’s Rule : ( ) ( ) ( ) ( ) ( ) ( ) ( )n n nùûVisit http://tutorial.math.lamar.edu for a complete set of Calculus notes.© 2005 Paul Dawkins

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