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 
                                        

                                          	
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                           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,