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