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Introduction to Integrals

Problems for "Average Value and Second Fundamental Theorem"

Average Value and Second Fundamental Theorem

Methods of Calculating Integrals

Problem : Find the average value of f for f (x) = 3x 2 - 6x on [- 1, 1] .


f avg   = f (x)dx  
    = x 3-3x 2 -1 1  
    = 1-3-(-1-3)  
    = 1  

Problem : Find the average value of f for f (x) = sin(x) on [- Π, Π] .


f avg   = f (x)dx  
    = -cos(x) -Π Π  
    = --1-(-(-1))  
    = 0  

Problem : For the function above, find the c on [- Π, Π] such that f (c) = f avg

sin(x) = 0 at x = 0 .

Problem : Use the second fundamental theorem of calculus to find F' .

F(x) = (9t+9)dt    

F'(x) = 9x + 9

Problem : Now integrate the above expression for F and then take the derivative to find F'(x) and verify the result from the previous question.

F(x) = (9t+9)dt    


9t+9 dt = t 2+9t 2 x      

x 2+9x - (18 + 18)
     
F(x) = x 2 +9x - 36      
F'(x) = 9x + 9      

This verifies the result given by the second fundamental theorem.

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