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Introduction to Integrals

Math
Summary

Problems for "Average Value and Second Fundamental Theorem" 4

Summary Problems for "Average Value and Second Fundamental Theorem" 4

Problem : Find the average value of f for f (x) = 3x2 - 6x on [- 1, 1].


favg = f (x)dx  
  = x3-3x2-11  
  = 1-3-(-1-3)  
  = 1  

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


favg = f (x)dx  
  = -cos(x)-ΠΠ  
  = --1-(-(-1))  
  = 0  

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

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+9dt = t2+9t2x    

x2+9x - (18 + 18)
    
F(x) = x2 +9x - 36    
F'(x) = 9x + 9    

This verifies the result given by the second fundamental theorem.