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4.3 Area and the Definite Integral

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4.3 Area and the Definite Integral Powered By Docstoc
					             4.3
Area and the Definite Integral

              "I have never let my
              schooling interfere with
              my education." -- Mark
              Twain
Objective…

   To evaluate the area under the curve of a
    figure
Integration story… part 1

   Two mathematicians were having dinner. One was
    complaining, "The average person is a mathematical
    idiot. People cannot do arithmetic properly, cannot
    balance a check book, cannot figure tips, cannot do
    percents,..." The other mathematician disagreed,
    "You're exaggerating. People know all the math they
    need to know." Later in the dinner, the complainer
    went to the men's room. The other mathematician
    beckoned to the waitress and said, "The next time
    you come past our table, I am going to stop you and
    ask you a question. No matter what I say, I want you
    to answer by saying "x-squared."
Integration story… part 2

   When the other mathematician returned, his
    companion said, "I am tired of your complaining. I
    am going to stop the next person who comes by our
    table and ask him or her an elementary calculus
    question, and I bet the person can solve it." Soon the
    waitress came by and he asked, "Excuse me, Miss,
    but can you tell me what the integral of 2x with
    respect to x is?" The waitress replied, "x-squared."
    The mathematician said, "See!" His friend said, "Oh I
    guess you're right." And the waitress said, "Plus a
    constant."
Derivatives and Integrals

   Derivative: Slope of the tangent line

   Integral: Area under the curve
Continuity implies Integrability

   Thm: If a function f is continuous on the
    closed interval [a,b], then it is integrable on
    [a, b]
Definite integral.. notation


   b

   
   a
       f ( x)dx 
What is the area of…

                       4

                        f ( x)dx 
                       0
  2

          f(x)




      0          2            4
Examples…

    Hint… graph the
     function first!


             
2
      4  x dx 
           2

2
3

 4dx 
1
3

 ( x  2)dx 
0
2 special definite integrals

   If f is defined at x = a then
            a

             f ( x)dx  0
            a
   If f is integrable on [a,b] then
                        b
        a
    b
            f ( x)dx    f ( x)dx
                        a
Examples



 sin xdx 



 0

  ( x  2)dx 
 3
Additive Interval Property

 Given that a<c<b, then
 b               c         b

  f ( x)dx   f ( x)dx   f ( x)dx
 a               a         c
Other properties

  b                     b

 a
   kf ( x)dx  k  f ( x)dx
                        a


 b                          b            b

   f ( x)  g ( x)dx    f ( x)dx   g ( x)dx
 a                          a            a
Pg 272, #22

				
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