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Introduction to Integrals

Problems for "The Definite Integral"

The Definite Integral

Average Value and Second Fundamental Theorem

Problem : Evaluate: sin(x)dx .

-cos(x) 0 Π =-(-1)-(-1) = 2    

Problem : Evaluate: (x 2-5x)dx .

( x 3 - x 2) 0 1=( - ) - (0) =    

Problem : Evaluate: (4x 2+1)dx .

x 3 + x -1 0=(0) - ( -1) =    

Problem : What is the relationship between f (x)dx and f (x)dx?

Since


f (x)dx   = F(b) - F(a), and  
f (x)dx   = F(a) - F(b),  
f (x)dx   = - f (x)dx  

Problem : For the function graphed below, find


    f (x)dx  
    f (x)dx  
    f (x)dx  

Recall the interpretation of the definite integral as the signed area under the curve. Portions below the graph count as "negative area" even though such a thing could not exist geometrically.


f (x)dx   = 3 - 1 + 2 = 4  
f (x)dx   = 1  
f (x)dx   = 3 - 1 + 2 - 5 = - 1  

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