Transcript CHAPTER 4
CHAPTER 4
THE DEFINITE INTEGRAL
4.1 Introduction to Area
• Finding area of polygonal regions can be
accomplished using area formulas for
rectangles and triangles.
• Finding area bounded by a curve is more
challenging.
• Consider that the area inside a circle is the
same as the area of an inscribed n-gon
where n is infinitely large.
Adding infinitely many terms
together
• Summation notation simplifies representation
• Area under any curve can be found by summing
infinitely many rectangles fitting under the curve.
Area f (t1 )t1 f (t2 )t2 f (t3 )t3 ... f (tn )tn
n
Area f (ti )ti
i 1
4.2 The Definite Integral
• Riemann sum is the sum of the product of
all function values at an arbitrary point in
an interval times the length of the interval.
• Intervals may be of different lengths, the
point of evaluation could be any point in
the interval.
• To find an area, we must find the sum of
infinitely many rectangles, each getting
infinitely small.
Definition: Definite Integral
• Let f be a function that is defined on
the closed
interval
[a,b].
n
_
f ( x i )xi exists, we say f is
• If lim
P 0
b
i 1
integrable on [a,b]. Moreover,
f ( x ) dx
called the definite integral
a
(or Riemann integral) of f from a to be,
is then given as that limit.
Area under a curve
• The definite integral from a to b of f(x) gives the
signed area of the region trapped between the
curve, f(x), and the x-axis on that interval.
• The lower limit of integration is a and the upper
limit of integration is b.
• If f is bounded on [a,b] and continuous except
at a finite number of points, then f is integrable
on [a,b]. In particular, if f is continuous on the
whole interval [a,b], it is integrable on [a,b].
Functions that are always
integrable
• Polynomial functions
• Sin & cosine functions
• Rational functions, provided that [a,b]
contains no points where the denominator
is 0.
4.3 First Fundamental Theorem
• Let f be continous on the closed
interval [a,b] and let x be a (variable)
point in (a,b). Then
x
d
f (t ) dt f ( x)
dx a
What does this mean?
• The rate at which the area under the curve
of function, f(t), is changing at a point is
equal to the value of the function at that
point.
4.4 The 2nd Fundamental Theorem
of Calculus and the Method of
Substitution
• Let f be continuous (integrable) on [a,b],
and let F be any antiderivative of f on [a,b].
Then the definite integral is
b
a
f ( x)dx F (b) F (a )
Evaluate
4
x
2
x
2
x
3
dx
x
3
x
2
4
2
4
4
3
4
(2)
2
2
4 3(4)
(2) 3(2)
4
4
64 16 12 4 4 6 42
4
4
Substitution Rule for Indefinite
Integrals
• Let g be a differentiable function and suppose
that F is an antiderivative of f. Then
f ( g ( x)) g ' ( x)dx F ( g ( x)) C
What does this remind you of?
• It is the chain rule! (from differentiation)
• In this case, you have an integral with a
function and it’s derivative both present in
the integrand.
• This is often referred to as “u-substitution”
• Let u=function and du=that function’s
derivative
Evaluate
2
cos
(
3
x
)
sin(
3
x
)
dx
{u cos(3x), du 3 sin(3x)dx}
1
2
(cos(3x)) 3 sin(3 x)dx
3
3
3
1 2
u
cos (3x)
u
du
C
C
3
9
9
Substitution Rule for Definite
Integrals
• Let g have a continuous derivative on
[a,b], and let f be continuous on the range
of g. Then where u=g(x):
g (b )
b
a
f ( g ( x)) g ' ( x)dx
f (u )du
g (a)
What does this mean?
• For a definite integral, when a substitution for u
is made, the upper and lower limits of
integration must change. They were stated in
terms of x, they must be changed to be the
corresponding values, in terms of u.
• When this change in the upper & lower limits is
made, there is no need to change the function
back to be in terms of x. It is evaluated in
terms of the upper & lower limits in terms of u.
Evaluate:
9
1
sin x
1
dx, u x , du
dx, u1 1 1, u2 9 3
2 x
2 x
3
sin udu ( cosu)
3
1
1
(cos3 cos1) 1.53
4.5 The Mean Value Theorem for
Integrals and the Use of Symmetry
• Average Value of a Function: If f is
integrable on the interval [a,b], then the
average value of f on [a,b] is:
b
f ave
1
f
(
x
)
dx
ba a
What does this mean?
• If you consider the definite integral from
over [a,b] to be the area between the
curve f(x) and the x-axis, f-average is the
height of the rectangle that would be
formed over that same interval containing
precisely the same area.
Mean Value Theorem for Integrals
• If f is continuous on [a,b], then there is a
number c between a and b such that
b
1
f (c )
f
(
t
)
dt
ba a
Symmetry Theorem
• If f is an even function then
a
a
a
0
f ( x)dx 2 f ( x)dx
• If f is an odd function, then
a
a
f ( x)dx 0
4.6 Numerical Integration
• If f is continuous on a closed interval [a,b],
then the definite integral must exist.
However, it is not always easy or possible
to find the definite integral.
• In these cases, we use other methods to
closely approximate the definite integral.
Methods for approximating a
definite integral
• Left (or right or midpoint) Riemann sums
(estimate the area with rectangles)
• Trapezoidal Rule (estimate with several
trapezoids)
• Simpson’s Rule (estimate the area with
the region contained under several
parabolas)
Summary of numerical techniques
• Approximating the definite integral of f(x) over
the interval from
a
to
b.
n
ba
ba
Riem ann:
f (a (i 1)
)
n i 1
n
b a n f ( xi 1 ) f ( xi )
Trapezoid :
2n i 1
2
Sim pson' s :
ba
[ f ( x0 ) 4 f ( x1 ) 2 f ( x2 ) 4 f ( x3 )...4 f ( xn 1 ) f ( xn )]
3n