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EGR 1101: Unit 10 Lecture #1
The Integral
(Sections 9.1, 9.2 of Rattan/Klingbeil text)
Antiderivative
Suppose f(x) is the derivative of F(x). Then F(x) is an antiderivative of f(x).
Example: 3x2 is the derivative of x3, so
x3 is an antiderivative of 3x2
A Function Has Many Antiderivatives
3x2 has many antiderivatives. One of them is x3. Another one is x3 + 1. Another one is x3 + 2. In fact, 3x2 has infinitely many
antiderivatives of the form x3 + C, where C is any constant.
The dx symbol
The symbol for antiderivative is dx. Read this as “antiderivative with respect to x.”
For example, we write
This means exactly the same thing as
Cxdxx 323
23 3)( xCxdx
d
Indefinite Integral
Another name for antiderivative is indefinite integral.
So we can also read
as “The indefinite integral with respect to x of 3x2 is x3 + C.”
Cxdxx 323
Table of Integrals
Just as we use a table of derivatives to differentiate functions, we use a table of integrals to integrate functions.
Many of the entries in a table of integrals are just the “reverse” of corresponding entries in a table of derivatives.
Differentiation and Indefinite Integration Cancel Each Other
Differentiation and indefinite integration are inverse operations, which means they cancel each other.
So
and Cxfdxdx
xdf )(
)(
)()( xfdxxfdx
d
Today’s Examples
1. Paving a driveway
2. Work
Definite Integration
The definite integral of a function f(x) from a to b is the area under the graph of that function between x=a and x=b.
The symbol for definite integration is
dxxfxxfAreab
a
n
ii
n
)()(lim
1
b
a
dx
Connection Between Definite Integration and Antiderivative
According to the Fundamental Theorem of Calculus,
where F(x) is an antiderivative of f(x). We use the following shorthand notation:
)()()( aFbFdxxfb
a
bab
a
xFdxxf )()(
Review: A Little History
Seventeenth-century mathematicians faced at least four big problems that required new techniques:
1. Slope of a curve
2. Rates of change (such as velocity and acceleration)
3. Maxima and minima of functions
4. Area under a curve
Using MATLAB to Integrate the Hard Part of Example #1
>> syms x
>> int(sqrt(2500-(x-50)^2), 0, 50)
Using MATLAB to Plot the Curves in Example #2
>> fplot('2*x^2+3*x+4', [0 1 0 100])
>> hold on>> fplot('2*sin(pi/2*x)+3*cos(pi/2*x)', [0 1 0 100], 'g')
>> fplot('4*exp(pi*x)', [0 1 0 100], 'r')
EGR 1101: Unit 10 Lecture #2
Applications of Integrals in Statics
(Sections 9.3, 9.4 of Rattan/Klingbeil text)
Today’s Examples
1. Centroid of a right triangle
2. Distributed load on a beam
Centroid
An area’s centroid is the point located at the “weighted-average” position of all points in the area.
For objects of uniform density, the centroid is the same as the object’s center of mass.
Centroids of Simple 2D Shapes
For a 2D planar lamina (very thin, rigid sheet of wood, metal, plastic, etc.), the centroid (denoted G) is the point at which you can balance it on your fingertip.
Unweighted Average Position
For n discrete objects located in a plane at coordinates (x1, y1), (x2, y2), …, (xn, yn), the unweighted average position is:
n
yy
n
xx
n
ii
n
ii
11 ,
Weighted Average Position
For n discrete objects located in a plane at coordinates (x1, y1), (x2, y2), …, (xn, yn), with weights p1, p2, … pn, the weighted average position is:
n
ii
n
iii
n
ii
n
iii
p
pyy
p
pxx
1
1
1
1 ,
Position of Centroid
For the area under a curve y(x) from x=a to x=b, the coordinates of the area’s centroid are given by
dxxy
dxxy
y
dxxy
dxxyx
x b
a
b
ab
a
b
a
)(
))((21
,
)(
)( 2
Position of Centroid (Using y-axis)
For the area under a curve x(y) from y=a to y=b, the x and y coordinates of the area’s centroid are given by
dyyx
dyyxy
y
dyyx
dyyx
x b
a
b
ab
a
b
a
)(
)(
,
)(
))((21 2
Statically Equivalent Loads
Two loads on a beam are statically equivalent if
1. they exert the same downward force, and
2. they exert the same moment (tendency to rotate the beam) about any point.
Example of Two Statically Equivalent Loads
Case 1
Case 2
Same downward force and same moment (tendency to rotate) in both cases, so these are statically equivalent.
25 lb
50 lb
25 lb
Example of Two Loads That Are Not Statically Equivalent
Case 1
Case 2
Same downward force in both cases, but different moments (tendencies to rotate), so these are not statically equivalent.
50 lb
50 lb
Moment of a Force
The moment of a force about a point is defined as the magnitude of the force times its distance from the point.
On previous slide, moment about the center point is zero in Case 2, but is nonzero in Case 1.
Finding Statically Equivalent Load
Problem: For a distributed load described by load curve w(x), find the size R and the location l of a concentrated load that is statically equivalent to the distributed load.
Solution: R is the area under the load curve . l is the x-coordinate of the centroid of the area
under the load curve.
Static Equilibrium(Again)
In Unit 4 we saw that for a system in static equilibrium, the external forces acting on the object add to zero:
The other condition required for static equilibrium is that the moments of the external forces about any point add to zero:
0
0
y
x
F
F
00 M