Download 5.2 - Hill City SD 51-2

5.2
Definite Integrals
What you’ll learn about
 Riemann Sums
 The Definite Integral
 Computing Definite Integrals on a Calculator
 Integrability
… and why
The definite integral is the basis of integral
calculus, just as the derivative is the basis of
differential calculus.
Sigma Notation
 a  a a  a  ...  a  a
n
k 1
k
1
2
3
n 1
n
The Definite Integral as a Limit of
Riemann Sums
Let f be a function defined on a closed interval [a, b]. For any partition P
of [a, b], let the numbers ck be chosen arbitrarily in the subinterval [ xk -1 , xk ].
n
If there exists a number I such that lim
 f (ck ) xk  I
P 0
k 1
no matter how P and the ck 's are chosen, then f is integrable on [a, b] and
I is the definite integral of f over [a, b].
The Existence of Definite Integrals
All continuous functions are integrable. That is, if a function f is
continuous on an interval [a, b], then its definite integral over
[a, b] exists.
The Definite Integral of a Continuous
Function on [a,b]
Let f be continuous on [ a, b], and let [ a, b] be partitioned into n subintervals
of equal length x  (b - a) / n. Then the definite integral of f over [a, b] is
given by lim  f (c )x, where each c is chosen arbitrarily in the
n
n 
th
k 1
k subinterval.
k
k
The Definite Integral
 f ( x)dx
b
a
Example Using the Notation
The interval [-2, 4] is partitioned into n subintervals of equal length x  6 / n.
Let m denote the midpoint of the k subinterval. Express the limit
th
k
lim   3  m
n
n 
k
k 1
  2m  5 x as an integral.
2
k
lim   3  m
n
n 
k 1
k
  2m  5x    3x  2 x  5  dx
2
4
k
2
2
Area Under a Curve (as a Definite
Integral)
If y  f ( x) is nonnegative and integrable over a closed interval [a, b],
then the area under the curve y  f ( x ) from a to b is the integral
of f from a to b, A   f ( x)dx.
b
a
Area
Area=   f ( x)dx when f ( x)  0.
b
a
 f ( x) dx   area above the x-axis    area below the x -axis  .
b
a
The Integral of a Constant
If f ( x)  c, where c is a constant, on the interval [a, b], then
 f ( x)dx   cdx  c(b  a )
b
b
a
a
Example Using fnINT
Evaluate numerically.
fnINT(x sin x, x,-1,2)  2.04
 x sin xdx
2
-1