Area under a Curve-Using a Limit

Transcription

1 Area uder a Curve-Usig a it Sice lettig be a very large umber will result i a huge amout of work, the process ca be simplified by usig sigma otatio ad summatio formulas to create a Riema Sum The ext example demostrates this cocept EXAMPLE : Fid the area uder the curve of the fuctio x 8 over the iterval [0, ] by usig rectagles Step : Determie the width rectagle by fidig x b a 0 x Step : Use the REP formula x a k to determie the height h f ( a k) rectagle k k k REP a k 0 k f ( a k) f 8 Step : Fid the area rectagle f ( a k) k 6k f ( a k) 8 Step : Fid the total area of all rectagles A T k A T 6k k 6 6 ( ) ( ) 8 A T 0 8( ) So, the approximate area of rectagles uder the curve is 08/ The umber 8/ represets the amout of excess area whe usig rectagles If the area was 0-8/ it would represet the shortfall of area for rectagles It is easy to see that the larger is the smaller the value of 8/ 8

2 To obtai a eve more accurate approximatio we should let the umber of rectagles approach ifiity This leads us to the last step i our procedure for fidig the area uder a curve which is to take the limit of the Riema Sum This is referred to as usig the limit process I the ext example the height rectagle will be calculated usig the right edpoit, although we could also use the left edpoit or midpoit as well EXAMPLE : Fid the area uder the curve of the fuctio [0, 8] usig the limit process 0 x over the iterval Step : Determie the width rectagle by fidig x b a x Step : Use the REP formula x a k to determie the height h f ( a k) rectagle REP 8 8k a k 0 k 8k 8k f ( a k) f 0 Step : Fid the area rectagle f ( a k) 8k 8 6k f ( a k) 0 Step : Fid the total area of all rectagles A T Area ( ) 6k 6k k 6 6 ( ) ( ) 8 Step 5: Fid the limit of the sum as approaches ifiity This will result i the actual area uder the curve Area 8 ( 8) The area uder the curve is 8 k

3 I the ext example the height rectagle will be calculated usig the left edpoit This makes the calculatios just a little trickier but as you will see, the results are the same EXAMPLE : Fid the area uder the curve of the fuctio [0, 8] usig the limit process 0 x over the iterval Step : Determie the width rectagle by fidig x b a x Step : Use the LEP formula x a ( k ) to determie the height h f ( a ( k ) rectagle LEP 8 8k 8 a ( k ) 0 ( k ), f ( a ( k )) 8k 8 8k 8 f 0 Step : Fid the area rectagle f ( a ( k )) 8k 8 8 6k 6 f ( a ( k )) 0 Step : Fid the total area of all rectagles A T k Area 6k 6 6k 6 k ( ) 6 ( ) ( ) ( ) Step 5: Fid the limit of the sum as approaches ifiity Area 8 ( 8) The area uder the curve is 8 I this last example whe usig the LEP formula we eded up with the limit of 8 / The / represets the excess area which resulted from usig circumscribed rectagles (Upper Sum) I the previous example whe usig the REP formula we eded up with the limit of 8 / The -/ represets the shortfall of area which resulted from usig iscribed rectagles (Lower Sum) As approaches ifiity, the quatity 8/ approaches zero

4 EXAMPLE : Fid the area uder the curve of the fuctio x over the iterval [0, 0] usig the limit process Step : Determie the width rectagle by fidig x b a x Step : Use the REP formula x a k to determie the height h f ( a k) rectagle 0 0k 0k 0k REP a k 0 k f ( a k) f Step : Fid the area rectagle f ( a ( k )) 0k 0 00k 0 f ( a k) Step : Fid the total area of all rectagles A T k Area 00k 0 00k 0 k ( ) 0 ( ) 50( ) Step 5: Fid the limit of the sum as approaches ifiity Area 60 ( 60) The area uder the curve is 60

5 EXAMPLE 5: Fid the area uder the curve of the fuctio [0, 5] usig the limit process x over the iterval Step : Determie the width rectagle by fidig x b a x Step : Use the REP formula x a k to determie the height h f ( a k) rectagle 5 5k 5k 5k REP a k 0 k f ( a k) f Step : Fid the area rectagle f ( a ( k )) 5k 5 5k f ( a xk) x Step : Fid the total area of all rectagles A T k 5k 5 Area ( k ) ( ) 5 ( ) Step 5: Fid the limit of the sum as approaches ifiity Area The area uder the curve is 5/ or 67

6 EXAMPLE 6: Fid the area uder the curve of the fuctio [, 5] usig the limit process x x over the iterval Step : Determie the width rectagle by fidig x b a 5 x Step : Use the REP formula x a k to determie the height h f ( a k) rectagle REP k k k k a k k f ( a k) f Step : Fid the area rectagle f ( a ( k )) k k 0 k 8k f ( a xk) x Step : Fid the total area of all rectagles A T k Area 0 k 8k 0 k 8 k k 0 ( ) ( ) ( ) Step 5: Fid the limit of the sum as approaches ifiity 8 ( ) Area The area uder the curve is 56/ or 867

7 EXAMPLE 7: Fid the area uder the curve of the fuctio [, ] usig the limit process x x over the iterval Step : Determie the width rectagle by fidig x b a x Step : Use the REP formula x a k to determie the height h f ( a k) rectagle k k k k REP a k k f ( a k) f Step : Fid the area rectagle f ( a ( k )) k k 6 k k k f ( a xk) x Step : Fid the total area of all rectagles A T Area k k k 6 6 ( ) k k ( ) k 6 k ( ) k ( ) 6 k Step 5: Fid the limit of the sum as approaches ifiity ( k k ) k 5 8 Area ( 8) The area uder the curve is 8