Math131 Calculus I The Definite Integral Notes 5.2
I. The
Definite Integral
If f is a function defined for a < x < b, we divide the interval [a,b] into n subintervals of equal
width 
. Let 
be the endpoints of the subintervals (note: a = 
and b = 
).
Let 
be any point in these subintervals. Then the definite
integral of f from a to b is:
as long as
the limit exists. (Note: we often take
the points to be the right endpoints)
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is called a Riemann Sum!
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Riemann Sum =
Riemann Sum =
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
In general, a definite integral can be thought of as a
__________ area!!
Math131 Calculus I Notes 5.2 page
2
ex#1 Use the
table below to find the lower and upper estimates for 
.
|
x |
0 |
5 |
10 |
15 |
20 |
25 |
|
f(x) |
-42 |
-37 |
-25 |
-6 |
15 |
36 |
ex#2 Evaluate
the Riemann Sum for 
. Use right endpoints,
a = 0, b = 3, and n= 6.
Math131 Calculus I Notes 5.2 page
3
II. The Midpoint
Rule
where 
and
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
ex#3 Use the
Midpoint Rule with n = 5 to approximate 
Math131 Calculus I Notes 5.2 page
4
III. Properties
of the Definite Integral
1.
2. 
3. 
4. 
5. 
6. 
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ex#4 If 
and 
, find 