Transcript Document
Homework questions thus far???
Section 4.10?
5.1?
5.2?
sin 2x
sin x dx
The Definite Integral
Chapters 7.7, 5.2 & 5.3
January 30, 2007
Estimating Area
vs
Exact Area
Pictures
Riemann sum rectangles, ∆t = 4 and n = 1:
Better Approximations
Trapezoid Rule uses straight lines
Trapezoidal Rule
Better Approximations
The Trapezoid Rule uses small lines
Next highest degree would be
parabolas…
Simpson’s Rule
Mmmm…
parabolas…
Put a parabola across each
pair of subintervals:
Simpson’s Rule
Mmmm…
parabolas…
Put a parabola across each
pair of subintervals:
So n must be even!
Simpson’s Rule Formula
Like trapezoidal
rule
Simpson’s Rule Formula
Divide by 3
instead of 2
Simpson’s Rule Formula
Interior
coefficients
alternate:
4,2,4,2,…,4
Simpson’s Rule Formula
Second from
start and end
are both 4
Simpson’s Rule
Uses Parabolas to fit the curve
b
a
x
f (x)dx
[ f (x0 ) 4 f (x1 ) 2 f (x2 ) 4 f (x3 ) ...
3
4 f (xn 1 ) f (xn )]
Where
n is even and
∆x = (b - a)/n
S2n=(Tn+
2Mn)/3
Use Simpson’s Rule to Approximate the
definite integral with n = 4
g(x) = ln[x]/x on the interval [3,11]
Use T4.
Runners:
A radar gun was used
to record the speed of
a runner during the first
5 seconds of a race
(see table) Use
Simpsons rule to
estimate the distance
the runner covered
during those 5
seconds.
t(s)
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
v(m/s)
0
4.67
7.34
8.86
9.73
10.22
10.51
10.67
10.76
10.81
10.81
Definition of Definite Integral:
If f is a continuous function defined for a≤x≤b, we
divide the interval [a,b] into n subintervals of equal
width ∆x=(b-a)/n. We let x0(=a),x1,x2,…,xn(=b) be the
endpoints of these subintervals and we let x1*, x2*, …
xn* be any sample points in these subintervals so
xi*lies in the ith subinterval [xi-1,xi]. Then the Definite
Integral of f from a to b is:
b
n
f (x)dx lim f (x )x
a
n
*
i
i 1
Express the limit as a Definite Integral
1 4ni
n
e
4
lim
n
n
4i
i 1 2
n
7
7i
7i
lim
2 3 3
n
n
n
i 1 n
n
2
Express the Definite Integral as a limit
2
(2 x
2
)dx
0
5
tan 2x dx
1
Properties of the Definite Integral
Properties of the Definite Integral
Properties of the Definite Integral
Properties of the Integral
b
a
a
b
f (x)dx f (x)dx
1)
a
f (x)dx
2)
=0
a
b
3)
b
cf (x)dx c f (x)dx
a
a
for “c” a constant
Properties of the Definite Integral
Given that:
1
2 f (x)dx 8
2
Evaluate the following:
1
f (x)dx ?
4
4
f (x)dx 3
1
2
g(x)dx 5
2
4
g(x)dx 7
2
1
f (x)dx ?
2
1
3dx ?
1
Properties of the Definite Integral
Given that:
Evaluate the following:
1
2 f (x)dx 8
2
4
f (x)dx 3
1
2
g(x)dx 5
2
4
g(x)dx 7
2
4
[3 f (x) 2g(x)]dx ?
2
2
3g(x)dx ?
2
4
f (x)dx
1
Given the graph of f, find:
Evaluate:
3
1
f (x)dx
1 x2
f (x) 1
2 x
1 x 0
0 x 1
1 x 3
Integral Defined Functions
Let f be continuous. Pick a constant a.
Define:
x
F(x)
f (t)dt
a
Integral Defined Functions
Let f be continuous. Pick a constant a.
Define:
x
F(x)
f (t)dt
a
Notes:
• lower limit a is a constant.
Integral Defined Functions
Let f be continuous. Pick a constant a.
x
Define:
F(x)
f (t)dt
a
Notes:
• lower limit a is a constant.
• Variable is x: describes how far to integrate.
Integral Defined Functions
Let f be continuous. Pick a constant a.
x
Define:
F(x)
f (t)dt
a
Notes:
• lower limit a is a constant.
• Variable is x: describes how far to integrate.
• t is called a dummy variable; it’s a placeholder
Integral Defined Functions
Let f be continuous. Pick a constant a.
x
Define:
F(x)
f (t)dt
a
Notes:
• lower limit a is a constant.
• Variable is x: describes how far to integrate.
• t is called a dummy variable; it’s a placeholder
• F describes how much area is under the curve up to x.
Example
x
Let f (x) 2 x . Let a = 1, and F(x)
f (t)dt .
Estimate F(2) and F(3).
a
x
F(x)
2 tdt
1
2
F(2)
1
2 tdt
1/ 2
f (1) 4 f (1.5) f (2)
3
1.8692
Example
x
Let f (x) 2 x . Let a = 1, and F(x)
f (t)dt .
Estimate F(2) and F(3).
a
x
F(x)
2 tdt
1
3
F(3)
2 tdt
1
1/ 2
f (1) 4 f (1.5) 2 f (2) 4 f (2.5) f (3)
3
1.8692
x
Where is F(x)
f (t)dt
increasing and decreasing?
a
f (t )
is given by the graph below:
F is increasing.
(adding area)
F is decreasing.
(Subtracting area)
Fundamental Theorem I
Derivatives of integrals:
Fundamental Theorem of Calculus, Version I:
If f is continuous on an interval, and a a number on that
interval, then the function F(x) defined by
x
F(x)
f (t)dt
a
has derivative f(x); that is, F'(x) = f(x).
Example
x
Suppose we define F(x)
2
cos(t
)dt .
2.5
Example
x
Suppose we define F(x)
2
cos(t
)dt .
2.5
Then F'(x) = cos(x2).
Example
x
2
F(x)
(t
Suppose we define
2t 1)dt .
7
Then F'(x) =
Example
x
2
F(x)
(t
Suppose we define
2t 1)dt .
7
Then F'(x) = x2 + 2x + 1.
Examples:
x
d
sin(t)dt
dx
2
y
d
2
d
2
5x dx 5x dx
dy y
dy 2
d
dr
r
cost dt
Examples:
d 2r
3
tant dt
dr
d 2
x dx
d 0
3
d
F[g(x)] F '[g(x)]g'(x)
dx
Fundamental Theorem of Calculus (Part 1)
If f is continuous on [a, b], then the function
defined by
x
F(x)
f (t)dt
axb
a
is continuous on [a, b] and differentiable on (a, b)
and
F '(x) f (x)
Fundamental Theorem of Calculus (Part 1)
(Chain Rule)
If f is continuous on [a, b], then the function
defined by
u( x )
F(x)
f (t)dt a x b
a
is continuous on [a, b] and differentiable on (a, b)
and
F '(x) f (u(x))u '(x)
In-class Assignment
1.
Find:
d cos x
ln t dt
1
dx 2
2.
a.
b.
c.
Estimate (by counting the squares) the total area between f(x) and the xaxis.
8
Using the given graph, estimate 0 f (x)dx
Why are your answers in parts (a) and (b) different?
Consider the function f(x) = x+1 on the interval [0,3]
First let the bottom bound = 1, if x >1, we calculate
the area using the formula for trapezoids:
1
b1 b2 h
2
Consider the function f(x) = x+1 on the interval [0,3]
Now calculate with bottom bound = 1, and x < 1, :
Consider the function f(x) = x+1 on the interval [0,3]
So, on [0,3], we have that
1 2
F(x) x 2x 3
2
And F’(x) = x + 1 = f(x)
as the theorem claimed!
Very Powerful!
Every continuous function is the derivative of some
x
other function! Namely:
f (t)dt
a