The Definite Integral

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The Definite Integral

• Dr. Ching I Chen

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5.1 Estimating with Finite Sums (1)Distance
Traveled
A train moves along a track at a steady rate of
75 miles per hours from 700 A.M. to 900 A.M.
What is the total distance traveled by the train?
Distance rate ? time area under the v-t
curve
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5.1 Estimating with Finite Sums (2)Distance
Traveled
The same story, if the velocity of the train is
various, What is the distance for a period of
time a, b
Distance area under the v-t curve
How to find the area ?
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5.1 Estimating with Finite Sums (3)Distance
Traveled
The general ideas to the area under a curve
between a domain a, b is to partitioned into
vertical strips
If the strips are narrow enough, they are almost
indistinguishable from rectangles. The sum of
the areas of these rectangles will give the
total area.
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5.1 Estimating with Finite Sums (4, Ex.
1-1)Distance Traveled
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5.1 Estimating with Finite Sums (5, Ex.
1-2)Distance Traveled
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5.1 Estimating with Finite Sums (6)Rectangular
Approximation Method (RAM)
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5.1 Estimating with Finite Sums (7)Rectangular
Approximation Method (RAM)
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5.1 Estimating with Finite Sums (8)Rectangular
Approximation Method (RAM)
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5.1 Estimating with Finite Sums (9)Rectangular
Approximation Method (RAM)
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5.1 Estimating with Finite Sums (10)Rectangular
Approximation Method (RAM)
All three sums approach the same number
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5.1 Estimating with Finite Sums (11, Ex
2)Rectangular Approximation Method (RAM)
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5.1 Estimating with Finite Sums (12, Exp.
1-1)Rectangular Approximation Method (RAM)
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5.1 Estimating with Finite Sums (13, Exp.
1-2)Rectangular Approximation Method (RAM)
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5.1 Estimating with Finite Sums (14, Exp.
1-3)Rectangular Approximation Method (RAM)
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5.1 Estimating with Finite Sums (15, Exp.
1-4)Rectangular Approximation Method (RAM)
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5.1 Estimating with Finite Sums (16, Ex.
3-1)Volume of a Sphere
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5.1 Estimating with Finite Sums (17, Ex.
3-2)Volume of a Sphere
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5.1 Estimating with Finite Sums (18)Cardiac
Output
Application to human physiology The number of
liters of blood your heart pumps in a fixed time
interval is called your cardiac output. How can
a physician measure a patients cardiac output
without interrupting the flow of blood ? One
technique is to inject a dye into a main vein
near the Heart. The dye is drawn into the right
side of the heart and pumped through the lungs
and out the left side of the heart into the
aorta.
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5.1 Estimating with Finite Sums (19)Cardiac
Output
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5.1 Estimating with Finite Sums (20, Ex.
4)Cardiac Output
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5.1 Estimating with Finite Sums (20, Ex.
4)Exercise
1, 2,3,4, 13, 20
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5.2 Definite Integrals (1) Riemann Sums
The sums in which we will be interested are
called Riemann sums
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5.2 Definite Integrals(2)Riemann Sums
Consider a function in a domain a, b ,
partition the interval a, b into n subintervals
by choosing n-1 points, say, x1, x2, , xn-1,
between a and b subject only to the
condition that a lt x1 lt x2 lt
lt xn-1 lt b
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5.2 Definite Integrals (3) Riemann Sums
a lt x1 lt x2 lt lt xn-1 lt b
Each subinterval, the area f(ck)?Dxk
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5.2 Definite Integrals (4) Riemann Sums
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5.2 Definite Integrals(5) Riemann Sums
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5.2 Definite Integrals (6, Theorem 1) Riemann
Sums
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5.2 Definite Integrals (7) Riemann Sums
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5.2 Definite Integrals (8) Terminology and
Notation of Integration
Integral of f from a to b
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5.2 Definite Integrals (9, Example 1)
Terminology and Notation of Integration
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5.2 Definite Integrals (10) Definite Integral
and Area
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5.2 Definite Integrals (11, Example 2) Definite
Integral and Area
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5.2 Definite Integrals (12) Definite Integral
and Area
If an integrable function y f(x) is
nonpositive, the nonzero terms f(ck)Dxk in the
Riemann sums for f over an interval a, b are
negatives. Then we have
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5.2 Definite Integrals (13) Definite Integral
and Area
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5.2 Definite Integrals (14, Exploration 1-15)
Definite Integral and Area
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5.2 Definite Integrals (15, Exploration 1-510)
Definite Integral and Area
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5.2 Definite Integrals (16, Theorem 2) Constant
Functions
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5.2 Definite Integrals (17, Example 3) Constant
Functions
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5.2 Definite Integrals (18, Example 4-a)
Integrals on a Calculator
syms x fn xsin(x) intfn int(fn, x, -1,
2) double(intfn) ?2.04276
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5.2 Definite Integrals (19, Example 4-b)
Integrals on a Calculator
syms x fn 4/(1xx) intfn int(fn, x, 0,
1) double(intfn) ?3.1416
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5.2 Definite Integrals (20, Example 4-c)
Integrals on a Calculator
syms x fn exp(-xx) intfn int(fn, x, 0,
5) double(intfn) ?0.88623
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5.2 Definite Integrals (21, Example 5)
Discontinuous Integrable Functions
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5.2 Definite Integrals (22, Exploration 2-1)
Discontinuous Integrable Functions
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5.2 Definite Integrals (23, Exploration 2-2)
Discontinuous Integrable Functions
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5.2 Definite Integrals (24, Exploration 2-3)
Discontinuous Integrable Functions
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5.2 Definite Integrals (25) Exercise
Exercise 13,15,17,19, 2932, 4346
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5.3 Definite Integrals and Antiderivatives (1)
Properties of Definite Integrals
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5.3 Definite Integrals and Antiderivatives (2)
Properties of Definite Integrals
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5.3 Definite Integrals and Antiderivatives (3)
Properties of Definite Integrals (Example 1)
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5.3 Definite Integrals and Antiderivatives (4)
Average Value of a Function (Example 2)
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5.3 Definite Integrals and Antiderivatives (5)
Average Value of a Function
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5.3 Definite Integrals and Antiderivatives (6)
Average Value of a Function (Example 3)
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5.3 Definite Integrals and Antiderivatives (7)
Mean Value Theorem for Definite Integrals
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5.3 Definite Integrals and Antiderivatives (8)
Mean Value Theorem for Definite Integrals
(Exploration 1-1,2)
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5.3 Definite Integrals and Antiderivatives (9)
Mean Value Theorem for Definite Integrals
(Exploration 1-3)
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5.3 Definite Integrals and Antiderivatives (10)
Connecting Differential and Integral Calculus
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5.3 Definite Integrals and Antiderivatives (11)
Connecting Differential and Integral Calculus
(Exploration 2-14)
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5.3 Definite Integrals and Antiderivatives (12)
Connecting Differential and Integral Calculus
(Exploration 2-5)
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5.3 Definite Integrals and Antiderivatives (13)
Connecting Differential and Integral Calculus
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5.3 Definite Integrals and Antiderivatives (14)
Connecting Differential and Integral Calculus
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5.3 Definite Integrals and Antiderivatives (15)
Connecting Differential and Integral Calculus
(Example 4)
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5.4 Fundamental Theorem of Calculus (1)
Fundamental Theorem, Part 1 (Theorem 4)
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5.4 Fundamental Theorem of Calculus (2)
Fundamental Theorem, Part 1 (Example 1)
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5.4 Fundamental Theorem of Calculus (3)
Fundamental Theorem, Part 1 (Example 2)
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5.4 Fundamental Theorem of Calculus (4)
Fundamental Theorem, Part 1 (Example 3)
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5.4 Fundamental Theorem of Calculus (5)
Fundamental Theorem, Part 1 (Example 4)
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5.4 Fundamental Theorem of Calculus (6)
Fundamental Theorem, Part 2 (Theorem 4)
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5.4 Fundamental Theorem of Calculus (7)
Fundamental Theorem, Part 2 (Example 5)
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5.4 Fundamental Theorem of Calculus (8) Area
Connection (Example 6)
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5.4 Fundamental Theorem of Calculus (9) Area
Connection
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5.4 Fundamental Theorem of Calculus (10) Area
Connection (Example 7)
With tool Matlab syms x fn xcos(2x) intfn
int(abs(fn), x, -3, 3) double(intfn)
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5.4 Fundamental Theorem of Calculus (11) Area
Connection
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5.4 Fundamental Theorem of Calculus (11) More
Applications (Example 8)
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5.4 Fundamental Theorem of Calculus (12) More
Applications
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5.4 Fundamental Theorem of Calculus (13) More
Applications (Example 9)
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5.5 Trapezoidal Rule (1) Trapezoidal
Approximations
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5.5 Trapezoidal Rule (2) Trapezoidal
Approximations
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5.5 Trapezoidal Rule (3, Example 1) Trapezoidal
Approximations
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5.5 Trapezoidal Rule (4, Example 2) Trapezoidal
Approximations
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5.5 Trapezoidal Rule (5) Other Algorithm
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5.5 Trapezoidal Rule (6, Example 3) Other
Algorithm
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5.5 Trapezoidal Rule (7) Error Analysis