# 4.3 Area and the Definite Integral by dfhdhdhdhjr

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```									             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
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

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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