MAT120 Asst. Prof. Ferhat PAKDAMAR (Civil Engineer) M Blok - M106 [email protected] Gebze Technical University Department of Architecture Spring – 2014/2015.

MAT120

Asst. Prof. Ferhat PAKDAMAR (Civil Engineer)

M Blok - M106

[email protected]

Gebze Technical UniversityDepartment of Architecture

Spring – 2014/2015

Week 10

SubjectsWeek Subjects Methods

1 11.02.2015 Introduction2 18.02.2015 Set Theory and Fuzzy Logic. Term Paper 

3 25.02.2015 Real Numbers, Complex numbers, Coordinate Systems.

4 04.03.2015 Functions, Linear equations  5 11.03.2015 Matrices  6 18.03.2015 Matrice operations  7 25.03.2015 MIDTERM EXAM MT

8 01.04.2015 Limit. Derivatives, Basic derivative rules

9 08.04.2015 Term Paper presentations Dead line for TP

10 15.04.2015 Integration by parts,  11 22.04.2015 Area and volume Integrals  12 29.04.2015 Introduction to Numeric Analysis  13 06.05.2015 Introduction to Statistics.  14 13.05.2015 Review15   Review  16   FINAL EXAM FINAL

INTEGRATION

Integration is a way of adding slices to find the whole.

Integration can be used to find areas, volumes, central points and many useful things. But it is easiest to start with finding the area under the curve of a function like this:

What is the area under y = f(x) ?

Integration

We could calculate the function at a few points and add up slices of width Δx like this (but the answer won't be very accurate):

We can make Δx a lot smaller and add up many small slices(answer is getting better): 

Slices

And as the slices approach zero in width, the answer approaches the true answer.

We now write dx to mean the Δx slices are approaching zero in width.

finding an Integral is the reverse of finding a Derivative.

Example: What is an integral of 2x?                                    

 We know that the derivative of x2 is 2x ...

 ... so an integral of 2x is x2

Notation

After the Integral Symbol we put the function we want to find the integral of (called the Integrand),and then finish with dx to mean the slices go in the x direction (and approach zero in width).

And here is how we write the answer:

The symbol for ‘Integral’ is a stylish ‘S’ (for ‘Sum’, the idea of summing slices):

Plus CWe wrote the answer as x2 but why + C ?

It is the "Constant of Integration". It is there because of all the functions whose derivative is 2x:

The derivative of x2+4 is 2x, and the derivative of x2+99 is also 2x, and so on! Because the derivative of a constant is zero.

So when we reverse the operation (to find the integral) we only know 2x, but there could have been a constant of any value.

So we wrap up the idea by just writing + C at the end.

Example: What is ∫x3 dx ?

On Rules of Integration there is a "Power Rule" that says:

∫xn dx = xn+1/(n+1) + C

We can use that rule with n=3:

∫x3 dx = x4 /4 + C

Definite vs Indefinite Integrals

We have been doing Indefinite Integrals so far.

A Definite Integral has actual values to calculate between (they are put at the bottom and top of the "S"):

Indefinite Integral Definite Integral

Common Functions Function Integral

Constant ∫a dx ax + C

Variable ∫x dx x2/2 + CSquare ∫x2 dx x3/3 + C

Reciprocal ∫(1/x) dx ln|x| + C

Exponential ∫ex dx ex + C

  ∫ax dx ax/ln(a) + C  ∫ln(x) dx x ln(x) − x + C

Trigonometry (x in radians) ∫cos(x) dx sin(x) + C

  ∫sin(x) dx -cos(x) + C  ∫sec2(x) dx tan(x) + C     

Integration Rules

Rules Function Integral

Multiplication by constant ∫cf(x) dx c∫f(x) dx

Power Rule (n≠-1) ∫xn dx xn+1/(n+1) + C

Sum Rule ∫(f + g) dx ∫f dx + ∫g dx

Difference Rule ∫(f - g) dx ∫f dx - ∫g dx

Example: what is the integral of sin(x) ?

From the table above it is listed as being −cos(x) + C

It is written as:

∫sin(x) dx = −cos(x) + C

Example: What is ∫√x dx ?

√x is also x0,5

We can use the Power Rule, where n=½:

∫xn dx = xn+1/(n+1) + C

∫x0,5 dx = x1,5/1,5 + C

Example: What is ∫cos x + x dx ?

Use the Sum Rule:

∫cos x + x dx = ∫cos x dx + ∫x dx

Work out the integral of each (using table above):

= sin x + x2/2 + C

Integration by Parts

Integration by Parts is a special method of integration that is often useful when two functions are multiplied together, but is also helpful in other ways.

You will see plenty of examples soon, but first let us see the rule:

∫u v dx = u∫v dx −∫u' (∫v dx) dx

u is the function u(x)v is the function v(x)

As a diagram:

Now put it together:

Example: What is ∫x cos(x) dx ?First choose u and v:

•u = x•v = cos(x)

Differentiate u: u' = x' = 1

Integrate v: ∫v dx = ∫cos(x) dx = sin(x)   

Simplify and solve:x sin(x) − ∫sin(x) dxx sin(x) + cos(x) + C

In English, to help you remember, ∫u v dx becomes:

(u integral v) minus integral of (derivative u, integral v)

So we followed these steps:

Choose u and v

Differentiate u: u‘

Integrate v: ∫v dx

Put u, u' and ∫v dx here: u∫v dx −∫u' (∫v dx) dx

Simplify and solve

Differentiate u: ln(x)' = 1/xIntegrate v: ∫1/x2 dx = ∫x-2 dx = −x-1 = -1/x   (by the power rule)

Example: What is ∫ln(x)/x2 dx ?

First choose u and v:•u = ln(x)•v = 1/x2

But there is only one function! How do we choose u and v ?

Simplify:−ln(x)/x − ∫−1/x2 dx = −ln(x)/x − 1/x + C−(ln(x) + 1)/x + C

Now put it together:

Choose u and v carefully!

Choose a u that gets simpler when you differentiate it and a v that doesn't get any more complicated when you integrate it.

A helpful rule of thumb is I LATE. Choose u based on which of these comes first:

I: Inverse trigonometric functions such as sin-1(x), cos-1(x), tan-

1(x)

L: Logarithmic functions such as ln(x), log(x)

A: Algebraic functions such as x2, x3

T: Trigonometric functions such as sin(x), cos(x), tan (x)

E: Exponential functions such as ex, 3x

Integration by Substitution"Integration by Substitution" (also called "u-substitution") is a method to find an integral, but only when it can be set up in a special way.

The first and most vital step is to be able to write our integral in this form:

Note that we have g(x) and its derivative g'(x)

Like in this example:

Here f=cos, and we have g=x2 and its derivative of 2xThis integral is good to go!

When our integral is set up like that, we can do this substitution:

Then we can integrate f(u), and finish by putting g(x) back as u.

Example: ∫cos(x2) 2x dx

We know (from above) that it is in the right form to do the substitution:

Now integrate:

∫cos(u) du = sin(u) + C

And finally put u=x2 back again:

sin(x2) + C

Example: ∫x/(x2+1) dx

Let me see ... the derivative of x2+1 is 2x ... so how about we rearrange it like this:

∫x/(x2+1) dx = ½∫2x/(x2+1) dx

Then we have:

½∫1/u du = ½ ln(u) + C

Now put u=x2+1 back again:½ ln(x2+1) + C

Then integrate:

Definite IntegralsA Definite Integral has start and end values: in other words there is an interval (a to b).

The values are put at the bottom and top of the "S", like this:

Indefinite Integral (no specific values)

Definite Integral (from a to b)

We can find the Definite Integral by calculating the Indefinite Integral at points a and b, then subtracting:

Example:

The Definite Integral, from 1 to 2, of 2x dx:

The Indefinite Integral is: ∫2x dx = x2 + C

At x=1: ∫2x dx = 12 + C

At x=2: ∫2x dx = 22 + C

Subtract:

(22 + C) − (12 + C)

22 + C − 12 − C

4 − 1 + C − C = 3

And "C" gets cancelled out ... so with Definite Integrals we can ignore C

In fact we can give the answer directly like this:

We can check that, by calculating the area of the shape:

Yes, it has an area of 3.

Example:

The Definite Integral, from 0,5 to 1,0, of cos(x) dx:

The Indefinite Integral is: ∫cos(x) dx = sin(x) + C

We can ignore C when we do the subtraction (as we saw above):

1

0,5

cos( ) sin(1) sin(0,5)

=0,841...-0,479....

=0,362...

x dx

PropertiesReversing the interval

Reversing the direction of the interval gives the negative of the original direction.

Interval of zero length

When the interval starts and ends at the same place, the result is zero:

Adding intervals

We can also add two adjacent intervals together:

Have a nice week!