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File: Integral Calculus Book Pdf Free Download 171320 | 11 Summary Integral
fundamental theorem of calculus riemann sums substitution integration methods 104003 differential and integral calculus i technion international school of engineering 2010 11 tutorial summary february 27 2011 kayla jacobs indefinite ...

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                                                                                     Fundamental Theorem of Calculus, 
                                                   Riemann Sums, Substitution Integration Methods 
                                                                                                            104003 Differential and Integral Calculus I 
                                                                                            Technion International School of Engineering 2010-11 
                                                                                            Tutorial Summary – February 27, 2011 – Kayla Jacobs 
                     
                    Indefinite vs. Definite Integrals 
                                                                                                                                                                                                                    A calc student upset as could be 
                     
                              ·         Indefinite integral:                                                                                                                                                  That his antiderivative didn't agree 
                                                                                                                                                                                                                   With the one in the book 
                                                            The function F(x) that answers question:                                                                                                                E'en aft one more look. 
                                                                                                                                                                                                                    Oh! Seems he forgot to write the "+ C". 
                                                               “What function, when differentiated, gives f(x)?”                                                                                                                        -Anonymous 
                                                                                                                                                                                                                     
                                                                                                 
                              ·         Definite integral:       
                                                                                              	
                                                  o  The number that represents the area under the curve f(x) between x=a and x=b 
                                                  o  a and b are called the limits of integration. 
                                                  o  Forget the +c. Now we’re calculating actual values . 
                     
                    Fundamental Theorem of Calculus  (Relationship between definite & indefinite integrals) 
                                                                                                                                                                                                                   
                    If 
    
 and f is continuous, then F is differentiable and 
    
. 
                                                    
                    Important Corollary: For any function F whose derivative is f   (i.e.,     ’     ), 
                                                                                                                                   &
                                                                                                                              % 
   
& 
 
                                                                                                                                 
                     
                    This lets you easily calculate definite integrals! 
                     
                    Definite Integral Properties 
                              ·             	  0 
                                        
                                          	
                              ·           	 
                                          	                                            
                              ·                                      whether or not   , 
                                           	                                     	                                    
                    Area in [a,b] Bounded by Curve f(x) 
                                                                                                                                                                               
                    Case 1: Curve entirely above x-axis. Really easy! Area =                                                                                                        
                                                                                                                                                                           
                                                                                                                                                                            	                             
                    Case 2: Curve entirely below x-axis. Easy! Area = |                                                                                             | = -                                 
                                                                                                                                                                                                      
                                                                                                                                                               	                                          	
                    Case 3: Curve sometimes below, sometimes above x-axis. Sort of easy! Break up into sections. 
                     
                    Average Value 
                    The average value of function f(x) in region [a,b] is:                                                                                            
                                                                                                                                      average   	                                               
              Calculus – Tutorial Summary – February 27,,  22001111                                                                                                                                        2 
               
              Riemann Sum 
               
              Let [a,b] ==  cclloosseedd  iinntteerrvvaall  iinn  tthhee  ddoommaaiinn  ooff  ffuunnccttiioonn  f 
               
              PPaarrttiittiioonn  [[aa,,bb]]  iinnttoo  nn  ssuubbddiivviissiioonnss::  {{  [[xx ,x ],  [x ,x ],  [x ,x ],  … , [x    ,x ]}  where a ==  xx  < x  < … < x                        < x  = b 
                                                                                  0    1        1    2        2    3              n-1    n                           0       1              n-1       n
               
              The Riemann sum ooff  ffuunnccttiioonn  ff  oovveerr  iinntteerrvvaall  [[aa,,bb]]  iiss:: 
                                                                                                  1           
                                                                                       *  + , ·   
                                                                                                 -20          -          -        -/0
                                                                                                                                                           th
              where y is any value between x  aanndd  xx.     Note (x – x ) is the length of the i  ssuubbddiivviissiioonn  [[xx                                                           , x]. 
                             i                                          i-1           i                   i      i-1                                                                   i-1     i
               
              If for all i:                                                      then… 
              y = x                                                              S = Left Riemann sum 
                i      i-1 
              y = x                                                              S = Right Riemann sum 
                i      i
              y = (x + x )/2                                                     S = Middle Riemann sum 
                i       i      i-1
              f(yi) = ( f(x         ) + f(x) )/2                                 S = Trapezoidal Riemann sum 
                                 i-1          i
              f(y) = maximum of f over [x , x]                                   S = Upper Riemann sum 
                   i                                            i-1    i
              f(y) = minimum of f over [x , x]                                   S = Lower Riemann sum 
                   i                                           i-1     i
                                                                                                                                                                                      
              As ' ( ∞, S ccoonnvveerrggeess  ttoo  tthhee  vvaalluuee  ooff  tthhee  ddeeffiinniittee  iinntteeggrraall of f over [a,b]:   lim                         *      
                                                                                                                                                               1((6                	
              Ex: Riemann sum methods of f(x))  ==  x3 over interval [a, b]]  ==  [[00,,  22]]  uussiinngg  44  eeqquuaall  ssuubbddiivviissiioonnss  ooff  00..55  eeaacchh:: 
               
              (1) Left Riemann sum:                                                                               (2) Right Riemann sum: 
                                                                                                                                                                                                             
              (4) Middle Riemann sum:                                                                             (3) TTrraappeezzooiiddaall  RRiieemmaannnn  ssuumm:: 
                                                                                                                                                                                                             
               
         Calculus – Tutorial Summary – February 27, 2011                                                                           3 
          
         Integration Method: u-substitution 
          
                                               %7·7  %977 
                                                	                              9	
          
                               
         …where 7    7’      (because 7’   7/). 
          
         Notes: 
              ·   This is basically derivative chain rule in reverse. 
              ·   The hard part is figuring out what a good u is. 
              ·   If it’s a definite integral, don’t forget to change the limits of integration!  ( 7,     ( 7 
              ·   If it’s an indefinite integral, don’t forget to change back to the original variable at the end, and +c. 
          
          
         Basic Trigonometric Derivatives and Indefinite Integrals 
          
                                    
               
                                                                                          
                                       
               
                                                                                             
                                      
               
                                                                                            
                                        
                                                                                               
               
                                           
                                                                                                 
               
                                             
                                                                                                    
                                                          
                                                     From trigonometric identities and u-substitution: 
                                                      
                                                                                                
                                                                                                         
                                                                                                         
          
         Calculus – Tutorial Summary – February 27, 2011                                                                           4 
          
          
         Integration Method: Trigonometric Substitution for Rational    
                                               Functions of Sine and Cosines 
          
          
         To integrate a rational function of sin(x) and cos(x), try the substitution: 
          
                                                  @  tan                        2    @ 
                                                            2                       1@?
          
         Use the following trig identities to transform the function into a rational function of t:  
                                                                   2 tan           2@
                                                      sin               2           ? 
                                                                  1tan?2         1@
                                                                 1tan?CD           1@?
                                                                            2
                                                     cos   1tan?CD1@? 
                                                                            2
                                                                  2tanC2D            2@
                                                     tan   1tan?CD1@? 
                                                                              2
          
         Integration Method: Trigonometric Substitution 
          
             If the integral involves…                     … then substitute…                   … and use the trig identity… 
                          E       E                             7   · sinG                                    ?             ?
                         F                                                                         1sin G  cos G 
                          E       E                            7   · tanG                                     ?             ?
                         F                                                                         1tan G  sec G 
                          E       E                             7   · secG                              ?                   ?
                        F                                                                          sec G1 tan G 
          
         Steps: 
              1.  Notice that the integral involves one of the terms above. 
              2.  Substitute the appropriate u. Make sure to change the dx to a du (with relevant factor). 
              3.  Simplify the integral using the appropriate trig identity. 
              4.  Rewrite the new integral in terms of the original non-Ѳ variable 
                            (draw a reference right-triangle to help). 
              5.  Solve the (hopefully now much easier) integral, 
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...Fundamental theorem of calculus riemann sums substitution integration methods differential and integral i technion international school engineering tutorial summary february kayla jacobs indefinite vs definite integrals a calc student upset as could be that his antiderivative didn t agree with the one in book function f x answers question e en aft more look oh seems he forgot to write c what when differentiated gives anonymous o number represents area under curve between b are called limits forget now we re calculating actual values relationship if is continuous then differentiable important corollary for any whose derivative this lets you easily calculate properties whether or not bounded by case entirely above axis really easy below sometimes sort break up into sections average value region sum let cclloosseedd iinntteerrvvaall iinn tthhee ddoommaaiinn ooff ffuunnccttiioonn ppaarrttiittiioonn iinnttoo nn ssuubbddiivviissiioonnss where xx n ff oovveerr iiss th y aanndd note length ssu...

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