The document discusses notation and properties for definite integrals. It defines the definite integral from a to b of f(x) dx as the area under the curve of f(x) between the x-axis and the limits of a and b. It lists five properties of definite integrals: 1) the order of integration does not matter, 2) the integral from a to a of any function f(x) is equal to zero, 3) a constant can be pulled out of the integral, 4) integrals can be added or subtracted, and 5) a definite integral over an interval can be broken into a sum of integrals over subintervals.

2. 
Notation for the definite integral…
b
f(x) dx
a
◦
◦

b is upper limit of integration
a is lower limit of integration
This equals the area under a curve, above
the x-axis if…
1. f(x) is continuous
2. f(x) is nonnegative

3. b
A rea
f ( x ) d x w h e n f(x)
0
a
b
A rea
f ( x ) d x w h e n f(x)
0
a
Net Area:
b
f ( x ) d x = (a re a a b ove x-a xis) - (a re a b e low x-a xis)
a

4. If f ( x )
c on [a ,b ], th e n
b
c d x = c(b -a )
a

5. b
Syntax: To evaluate
f(x) dx …
a
MATH 9 fnInt(

6. 1. Order of Integration
a
b
f(x) dx
b
f(x) dx
a

7. 2. Zero
a
f(x) dx = 0
a

8. 3. Constant Multiple
b
b
k f(x) dx
a
k
f (x ) dx
a

9. 4. Sum and Difference
b
b
[ f(x)
a
g ( x )] d x
b
f(x) dx
a
g( x ) d x
a

10. 5. Additivity
b
c
f(x) dx +
a
c
f(x) dx
b
f (x) dx
a

11. a
b
1. Order of Integration
f(x) dx
f(x) dx
b
a
a
2. Zero
f(x) dx = 0
a
b
b
3. Constant Multiple
k f(x) dx
k
a
f (x ) dx
a
b
4. Sum and Difference
b
[ f(x)
g ( x )] d x
a
b
f(x) dx
a
b
5. Additivity
c
f(x) dx +
a
a
c
f(x) dx
b
g( x ) d x
f (x) dx
a