6.4 - The Definite Integral
I. The Definite Integral
If we let the number of rectangles on [a, b] increase without bound and if f is continuous on [a, b], then
n
lim
 f (ck ) x
n   k 1
is the definite integral of f (x) on [a, b] and we write
n
 f (ck ) x
lim
n  k 1

b
a f (x) dx.
Geometric interpretation:
II. Properties:
(in addition to all the previous properties of integrals)
a
1.
 f (x) dx  0
2.
a
3.
If f ( x)  0,
b
a
a
b
 f (x) dx  
 f (x) dx
b
 f (x) dx  0
a
4. If
a  c  b , then
b
c
a
a
 f (x) dx   f (x) dx

b
 f (x) dx .
c
Examples:
4
1.
 2 dx
1
3
2.
 ( 2x  1) dx
2
More generally,
(1) f can take on negative values as well as positive (2) rectangles can vary in width
(3) points to compute height can be anywhere in the subintervals
Riemann Sums: Sn 
n
 f (ck ) x
k 1
where ck  [ xk 1, xk ] .