Post on 01-Jan-2016
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Homework questions thus far???
Section 4.10? 5.1? 5.2?
sin 2x
sin xdx∫
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 RuleMmmm…parabolas…
Put a parabola across each pair of subintervals:
Simpson’s RuleMmmm…parabolas…
Put a parabola across each pair of subintervals:
So n must be even!
Simpson’s Rule Formula
Like trapezoidalrule
Simpson’s Rule Formula
Divide by 3instead of 2
Simpson’s Rule Formula
Interiorcoefficientsalternate:
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
f (x)dxa
b
∫ ≈Δx3[ f (x0 ) + 4 f (x1) + 2 f (x2 ) + 4 f (x3) + ...
+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) v(m/s)0 0
0.5 4.671 7.34
1.5 8.862 9.73
2.5 10.223 10.51
3.5 10.674 10.76
4.5 10.815 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:
f (x)dxa
b
∫ =limn→ ∞
f(xi* )
i=1
n
∑ Δx
Express the limit as a Definite Integral
limn→ ∞
e1+4 in
⎛⎝⎜
⎞⎠⎟
2 +4in
⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟i=1
n
∑ 4n
limn→ ∞
7n2 3+
7in
⎛⎝⎜
⎞⎠⎟+ 3+
7in
⎛⎝⎜
⎞⎠⎟
2
i=1
n
∑
Express the Definite Integral as a limit
(2−x2 )dx0
2
∫
tan 2x( )dx1
5
∫
Properties of the Definite Integral
Properties of the Definite Integral
Properties of the Definite Integral
Properties of the Integral
1) f (x)dx =a
b
∫ − f(x)dxb
a
∫
f (x)dxa
a
∫
cf (x)dx =c f(x)dxa
b
∫a
b
∫
2) = 0
3) for “c” a constant
Properties of the Definite Integral
Given that:
2 f (x)dx =8−2
1
∫
f(x)dx=31
4
∫
g(x)dx=2
−2
∫ 5
g(x)dx=2
4
∫ −7
Evaluate the following:
f (x)dx4
1
∫ =?
f (x)dx−2
1
∫ =?
3dx−1
1
∫ =?
Properties of the Definite Integral
Given that:
2 f (x)dx =8−2
1
∫
f(x)dx=31
4
∫
g(x)dx=2
−2
∫ 5
g(x)dx=2
4
∫ −7
Evaluate the following:
3g(x)dx−2
2
∫ =?
[3 f (x)−2g(x)]dx−2
4
∫ =?
Given the graph of f, find:
f (x)dx−1
4
∫
Evaluate:
f (x)dx−1
3
∫ f(x) =1−x2 −1≤x≤01 0 ≤x≤12 −x 1≤x≤3
⎧
⎨⎪
⎩⎪
Integral Defined Functions
Let f be continuous. Pick a constant a.
Define: F(x)= f(t)dta
x
∫
Integral Defined Functions
Let f be continuous. Pick a constant a.
Define:
Notes:
• lower limit a is a constant.
F(x)= f(t)dta
x
∫
Integral Defined Functions
Let f be continuous. Pick a constant a.
Define:
Notes:
• lower limit a is a constant.• Variable is x: describes how far to integrate.
F(x)= f (t)dta
x
∫
Integral Defined Functions
Let f be continuous. Pick a constant a.
Define:
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(x)= f (t)dta
x
∫
Integral Defined Functions
Let f be continuous. Pick a constant a.
Define:
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.
F(x)= f(t)dta
x
∫
Example
Let . Let a = 1, and .
Estimate F(2) and F(3).
F(x)= f (t)dta
x
∫f (x)= 2 + x
F(x)= 2 + tdt1
x
∫
F(2)= 2 + tdt1
2
∫ ≈1 / 2
3f (1)+ 4 f(1.5) + f(2)[ ] ≈
1.8692
Example
Let . Let a = 1, and .
Estimate F(2) and F(3).
F(x)= f (t)dta
x
∫f (x)= 2 + x
F(x)= 2 + tdt1
x
∫
F(3)= 2 + tdt1
3
∫≈
1 / 2
3f (1) + 4 f (1.5) + 2 f (2) + 4 f (2.5) + f (3)[ ]
≈1.8692
Where is increasing and decreasing?
is given by the graph below: f (t)F is increasing. (adding area)
F is decreasing.
(Subtracting area)
F(x)= f (t)dta
x
∫
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
has derivative f(x); that is, F'(x) = f(x).
F(x)= f(t)dta
x
∫
Example
Suppose we define .F(x)= cos(t2 )dt2.5
x
∫
Example
Suppose we define .
Then F'(x) = cos(x2).
F(x)= cos(t2 )dt2.5
x
∫
Example
Suppose we define .
Then F'(x) =
F(x)= (t2 + 2t+1)dt−7
x
∫
Example
Suppose we define .
Then F'(x) = x2 + 2x + 1.
F(x)= (t2 + 2t+1)dt−7
x
∫
Examples:
d
dxsin(t)
−π
x
∫ dt⎛
⎝⎜⎞
⎠⎟
d
dy5x2dx
y
2
∫⎛
⎝⎜
⎞
⎠⎟ =
d
dy− 5x2dx
2
y
∫⎛
⎝⎜
⎞
⎠⎟
d
drcos t dt
−π
r
∫
Examples:
d
dθx2dx
0
θ 3
∫⎛
⎝⎜
⎞
⎠⎟
d
drtan t 3 dt
−π
2r
∫d
drtan t 3 dt
−π
2r
∫
d
dθx2dx
0
θ 3
∫⎛
⎝⎜
⎞
⎠⎟
d
dxF[g(x)][ ] =F '[g(x)]g'(x)
If f is continuous on [a, b], then the function defined by
is continuous on [a, b] and differentiable on (a, b) and
Fundamental Theorem of Calculus (Part 1)
F(x)= f(t)dta
x
∫ a≤x≤b
F '(x)= f(x)
If f is continuous on [a, b], then the function defined by
is continuous on [a, b] and differentiable on (a, b) and
Fundamental Theorem of Calculus (Part 1)(Chain Rule)
F(x)= f(t)dta
u(x)
∫ a≤x≤b
F '(x)= f(u(x))u'(x)
In-class Assignment
a. Estimate (by counting the squares) the total area between f(x) and the x-axis.
b. Using the given graph, estimatec. Why are your answers in parts (a) and (b) different?
€
f (x)dx0
8
∫
d
dxln t dt1
2
cos x
∫⎛
⎝⎜⎞
⎠⎟
2.
1. Find:
First let the bottom bound = 1, if x >1, we calculate the area using the formula for trapezoids:
1
2b1 +b2( ) h( )
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]
Consider the function f(x) = x+1 on the interval [0,3]
So, on [0,3], we have that
And F’(x) = x + 1 = f(x) as the theorem claimed!
Very Powerful!Every continuous function is the derivative of some
other function! Namely:
F(x)=12x2 + 2x−3( )
f (t)dta
x
∫