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5.4 Definite Integrals and
          Areas
What’s a definite integral?
You know how to integrate a function and get an indefinite integral :

∫ f ( x ) dx ⇒ F ( x ) + C
You know that part before + C. Let' s call it capital F(x).
So here is the notation for a definite integral on an interval (a, b) :

∫ f ( x ) dx
 b

 a

And here is how you find it :
F ( b) − F ( a )
                 4       x   3 4
                                     ( 4 ) 3   ( 2 ) 3  64 8 56
EXAMPLE : ∫          x =
                      2
                                   =         −
                                     3   3  3 −3= 3
                                                          =
                 2       3     2                       
2
∫       x dx =
         2
1



    5
∫       x dx =
         2
2



                 −1
You try : ∫ x         2
                 −3
Find the area under y = e 2 x from x = 0 to x = 1.
 1
∫
0
     e 2 x dx =




You try : Find the area under y = e 2 x from x = 1 to x = 3.
Find the area under the curve f ( x ) = − x 2 + 3 over the interval - 1 to 1.




                                       1
Find the area under the curve f ( x ) = from x = 1 to x = e.
                                       x
You try : Find the area under the curve f ( x ) = 24 − x 2 over the interval - 1 to 1.




                                                    1
You try : Find the area under the curve f ( x ) =     from x = 1 to x = e 2 .
                                                    x
Can you have negative area? No.
If an area can' t be negative, then how come
    2
∫       -x =
         2
1

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125 5.4

  • 2. What’s a definite integral? You know how to integrate a function and get an indefinite integral : ∫ f ( x ) dx ⇒ F ( x ) + C You know that part before + C. Let' s call it capital F(x). So here is the notation for a definite integral on an interval (a, b) : ∫ f ( x ) dx b a And here is how you find it : F ( b) − F ( a ) 4 x 3 4  ( 4 ) 3   ( 2 ) 3  64 8 56 EXAMPLE : ∫ x = 2 = −  3   3  3 −3= 3 = 2 3 2    
  • 3. 2 ∫ x dx = 2 1 5 ∫ x dx = 2 2 −1 You try : ∫ x 2 −3
  • 4. Find the area under y = e 2 x from x = 0 to x = 1. 1 ∫ 0 e 2 x dx = You try : Find the area under y = e 2 x from x = 1 to x = 3.
  • 5. Find the area under the curve f ( x ) = − x 2 + 3 over the interval - 1 to 1. 1 Find the area under the curve f ( x ) = from x = 1 to x = e. x
  • 6. You try : Find the area under the curve f ( x ) = 24 − x 2 over the interval - 1 to 1. 1 You try : Find the area under the curve f ( x ) = from x = 1 to x = e 2 . x
  • 7. Can you have negative area? No. If an area can' t be negative, then how come 2 ∫ -x = 2 1