Transcript Fundamental theorem of calculus examples

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
 3x dx  x
2

3
C
This means exactly the same thing as
d 3
2
( x  C )  3x
dx
Indefinite Integral


Another name for antiderivative is
indefinite integral.
So we can also read
 3x dx  x
2
3
C
as “The indefinite integral with respect to x
of 3x2 is x3 + C.”
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
d
dx 
and
f ( x)dx  f ( x)
df ( x)
 dx dx  f ( x) C
Today’s Examples
1.
2.
Paving a driveway
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.
b

The symbol for definite integration is
 dx
a
n
b
i 1
a
Area  lim  f ( xi )x  f ( x)dx
n 
Connection Between Definite
Integration and Antiderivative

According to the Fundamental Theorem of
Calculus,
b
 f ( x)dx  F (b)  F (a)
a

where F(x) is an antiderivative of f(x).
We use the following shorthand notation:
b
 f ( x)dx  F ( x)
b
a
a
Review: A Little History

1.
2.
3.
4.
Seventeenth-century mathematicians
faced at least four big problems that
required new techniques:
Slope of a curve
Rates of change (such as velocity and
acceleration)
Maxima and minima of functions
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.
2.
Centroid of a right triangle
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
x
x
i 1
n
i
n
, y
y
i 1
n
i
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
x
x p
i 1
n
i
i
p
i 1
i
n
, y
y p
i 1
n
i
i
p
i 1
i
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
b
x
 x  y( x)dx
a
b
b
, y
1
2
( y ( x)) dx

2a
b
 y( x)dx
 y( x)dx
a
a
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
b
x
1
2
( x( y )) dy

2a
b
b
, y
 y  x( y)dy
a
b
 x( y)dy
 x( y)dy
a
a
Statically Equivalent Loads

Two loads on a beam are statically
equivalent if
1.
2.
they exert the same downward
force, and
they exert the same moment
(tendency to rotate the beam)
about any point.
Example of Two Statically Equivalent
Loads
25 lb
25 lb
Case 1
50 lb
Case 2

Same downward force and same moment
(tendency to rotate) in both cases, so
these are statically equivalent.
Example of Two Loads That Are Not
Statically Equivalent
50 lb
Case 1
50 lb
Case 2

Same downward force in both cases, but
different moments (tendencies to rotate),
so these are not statically equivalent.
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:
 Fx  0
F
y

0
The other condition required for static
equilibrium is that the moments of the
external forces about any point add to
zero:
 M0  0