Numerical methods for calculation of integrals

Publicerat den april 27, 2012 av mattelararen

For many functions it is impossible to find a primtive function and therefore it is impossible to use the fundamental theorem of calculus to solve the integral.

Luckily there are ways to cope with such circumstances with the aid of numerical methods. Here one has deviced a formula for approximation of the integral.

The perhaps the simplest one is the trapezoidal formula.

The principle is to divide the area into well-known entities( in this case trapezoids) for which the area may readily be computed.

A more refined method is thee.

 Simpson’s rule. This can be viewed as a combination of the midpoint-rule and the trapezoidrule. The midpoint rule basically amounts to approximating the area by rectangles. 

Publicerat i Calculus, matematik 5 | Märkt | Lämna en kommentar

Vector algebra-adding and subtacting vectors

Publicerat den april 21, 2012 av mattelararen

Garden

The mass of your body is a measure of the amount of matter (atoms) that constitute your body. This number is a quantity that can be pinpointed on the x-axis (or any line of numbers). Such quantities are called scalars. 

It is important here to bear in mind the difference between weight and mass.

The weight is the pull of gravity acting upon any mass within the gravitational field.  This quantity not only has a size but also a direction. It is directed towards the center of gravity of the earth.

Galaxy M 83 15 million light-years away held together by gravity. Courtesy: Anglo-Australian telescope.

Such quantities possessing both mass and direction are termed vectors.

Many important physical quantities are not just quantities (As eg mass, energy and temperature) but  they also have a direction (as e.g. velocities, forces, momentum, pressure, accelerations).

They are usually represented by the length and direction of an arrow or by the coordinates (x, y, z) of the endpoint of the arrow representing the vector beginning at the origin. They are often denoted by a bold letter (F) with an arrow above it.

Position vector

Vectors are added by the polygonal method which means that in order to obtain the vectorsum of several vectors you let the vector number two start at the endpoint of the first vector and so on. After you have drawn all the vectors like this after each other you are able to construct the vectorsum, or resultant, of all the vectors by drawing one vector from the startpoint of the first vector to the endpoint of the last vector.

File:Vectoraddition.svg

A vector can always be divided into x,y, and z-components

F=Fx i+ Fy j+ Fz k.

where i, j, k are the orthogonal unit vectors for the cartesian coordinatesystem.

This gives us the possibility to add vectors algebraically:

 F + G =  (Fx+Gx, Fy+Gy, Fz+Gz) i.e. add the x-coordinates separately and do likewise with the y and z-coordinates to acquire the coordinates of the sum.

Publicerat i Advanced | Märkt , | Lämna en kommentar

Symmetries, translations and dilatations

Publicerat den april 16, 2012 av mattelararen
A good definition of symmetry was given by the mathematician Hermann Weyl: 
An object is symmetrical  if, after you performed an operation on it, it still looks the same as it did before.
  • A function can be moved in the horizontal direction so-called translation, by adding or subtracting a number. For example :

f(x) = sin(x-a) is the function f(x) = sin(x) moved a steps to the right.

  • The graph of a function can be moved in vertical direction (vertical translation) by adding or subtracing a number to the function e.g. f(x) = sin(x) +4 is the function f(x) = sin(x) moved four steps upwa
  •  A vectorial translation i.e. a translation in both the x- and the y- direction.  y= sin (x+a) +b.
  • By dilatation is meant a contraction or comprimation of the function. f(x) = sin(x/m) is an extraction of the function f(x)=sin(x/m) by the scale m.
  • A function may also be mirrored in the x-axis. -f(x) is the mirror -image of f(x) in the x-axis. f(x)=-sin(x) is the mirror-image of f(x) = sin(x) in the x-axis.
  • f(-x) is the mirror-image in the y-axis of f(x). f(x)=sin(-x) mirrors f(x) = sin(x) in the y-axis.
  • By combining both types of mirroring one obtains a mirroring in the origin of the function.
Publicerat i Calculus, Gymnasiematematik(high school math), Uncategorized | Lämna en kommentar

Properties of integrals

Publicerat den april 11, 2012 av mattelararen
  1.  the integral of a linear combination is the linear combination of the integrals,
    \int_a^b (\alpha f + \beta g)(x) \, dx = \alpha \int_a^b f(x) \,dx + \beta \int_a^b g(x) \, dx. \,

If a > b then define

\int_a^b f(x) \, dx = - \int_b^a f(x) \, dx.

3.

Additivity of integration on intervals. If c is any element of [a, b], then \int_a^b f(x) \, dx = \int_a^c f(x) \, dx + \int_c^b f(x) \, dx.
4.
Upper and lower bounds. An integrable function f on [a, b], is necessarily bounded on that interval. Thus there are real numbers m and M so that m ≤ f (x) ≤ M for all x in [a, b]. Since the lower and upper sums of f over [a, b] are therefore bounded by, respectively, m(b − a) and
M(b − a), it follows that
      m(b - a) \leq \int_a^b f(x) \, dx \leq M(b - a).
 

20140507-183437.jpg

Applications in physics/technology of integrals.

Publicerat i Calculus, matematik 3c, matematik 4 | Märkt | Lämna en kommentar

Definite integrals

Publicerat den april 2, 2012 av mattelararen

In a geometrical context the integral can be interpreted as the area between the graph of the integrand f(x) and the x-axis. The idea is to summarize infinitely many infinitely thin rectangles. (An alternative approach to areacomputation is the Lebesgue-integration where you partition the range of f(x)  instead of the domain). This method works better for integration of infinite series e.g. Fourier series than the Riemann integral.

The integral sign is a stylized version of the letter ‘s’.

Bernhard Riemann discovered that this can be accomplished by forming one ‘uppersum’ and one ‘lowersum’.

An undersum can be seen here and to see the oversum click here.

Riemann showed that there exists exacctly one number which is smaller than the uppersum and bigger than the lower sum in the limit when the width of the rectangles approaches zero.

This number is identical to the integral over this interval.

Earlier  in the 18th. century Newton and Leibniz discovered the fundamental theorem of calculus. This gives an easy method for computation of the integral with the aid of the primitive function

F'(x) =f(x)  then       F(x) = \int_a^x f(t)\, dt\,.

See the derivation of this theorem here in an excerpt from  the Gamma textbook (björup et. al.) Fundamentaltheorem.

F'(x) = (F(x+dx) -F(x))/dx   F’ (x) dx = (F(x+h) – F(x)).

By Summation of  all these differential areaelements   it can be seen that all terms cancel each other except the first and the last which gives us the fundamental theorem of calculus:

F(b) - F(a) = \int_a^b f(x)\,dx,


pp.175-180

Ex
=
∫x2dx = x3/3 = 33/3 -03  /3= 27/3 = 9.

Publicerat i Calculus, Gymnasiematematik(high school math), matematik 3c, matematik 4 | Märkt | 1 kommentar

Indefinite integrals

Publicerat den mars 28, 2012 av mattelararen

If you need to calculate the distance travelled when you know the velocity as a function of time , since s'(t) = v(t) you need to be able to perform antiderivation i.e. finding a function whose derivative equals your function.

This process is called integration.

Hence s(t) is the indefinite integral or primitive function, to v(t).

For an arbitrary function f(x) the primitive funktion is conventionally written as

F = \int f(x)\,dx. 
and  F’ (x) =  f(x). 
Table of indefinita integrals: 
formula                         
formula
formula
formula

formula

formula

formula

formula

formula

formula

formula

formula

formula

formula

formula

formula

formula

formula

formula

formula

formula

formula

Integrazione per sostituzione

Per il calcolo di integrali del tipo formula, talvolta può essere vantaggioso sostituire alla variabile d’integrazione x una funzione di un’altra variabile t, purché tale funzione sia derivabile e invertibile.

Ponendo formula, da cui deriva formula, si ha che:

formula di integrazione per sostituzione

Read moreIndefinite integrals – primitiva funktioner

p. 161-169

Publicerat i Calculus, Gymnasiematematik(high school math), matematik 3c, matematik 4 | Märkt , | Lämna en kommentar

Differentiating the natural logarithm, products and quotients

Publicerat den mars 26, 2012 av mattelararen

In order to be able to deduce the derivative of the natural logarithm we resort to using implicit differentiation.

Let x= ey(x)

Differentiating both sides gives

dx/dx = d ey(x)/dx

1=ey(x) dy(x)/dx

Solving for dy(x)/dx one obtains

dy(x)/dx = 1/ey(x) = 1/x .

The product rule is given by

d f(x) g(x)/dx = df(x)/dxg(x) + f(x) dg(x)/dx

A beautiful proof for this theorem is given by G.W. Leibniz:

(u + du) *(v+dv) = u*v + u*dv + v* du + du*dv.

The last term is the product of two infinitesimals and can therefore be neglected. The differential of the product of the two functions u(x)*v(x) is thus equal to

u(x)*dv + v(x)*du.

Q.E.D.

To read more about this formula click here.

The rule for differentiation of the quotient of  two functions can be deduced from the product rule and is given by

df(x)/g(x) =( df(x)/dx g(x) – f(x) dg(x))/dx)/(g(x))2.

Publicerat i Calculus, Gymnasiematematik(high school math), matematik 3c, matematik 4 | Märkt , , | Lämna en kommentar

Pi-day

Publicerat den mars 16, 2012 av mattelararen

March the 14th. has officially been named the international π-day to honour this magical number which equals the ratio of the circmference to the diameter for all circles.

http://www.wikihow.com/Celebrate-Pi-Day

In 1882 the german mathematician Ferdinand Lindemann showed that pi is a transcendental number http://en.wikipedia.org/wiki/Transcendental_number

Such numbers have an infinitesimal decimal expansion with no periodicity in the numbers.

It has been said that the decimalexpansion of pi is the perfect generator of random numbers.

The record is ten million decimals generated with a computer using the Taylor-series expansion of arcustangens π/4 =1 and then solving for pi.

Perhaps the strangest quality of pi is that it surfaces in many areas of science other than geometry.

Probability calculus, imaginary numbers, infinitesimal series, calculus e.g..

So in the infinite number of decimals of pi are hidden the answers to many scientific enigmas. http://en.wikipedia.org/wiki/Pi

Publicerat i Geometri, Gymnasiematematik(high school math) | Lämna en kommentar

Differentiation of the trigonometric functions

Publicerat den mars 9, 2012 av mattelararen

To be able to differentiate the trigonometric functions one needs some standard limits:

With the aid of these and the definition of the derivative

f'(x_0)= \lim_{h \to 0} \frac{f(x_0+h) - f(x_0)}{h}

it is possible to show that

f(x)= sin (x) implies  f ‘(x) = cos(x)

and

f(x) = cos(x) implies f ‘(x) = -sin (x).

Publicerat i Calculus, Gymnasiematematik(high school math), matematik 4 | Märkt | Lämna en kommentar

Radians and an attempt at squaring the circle

Publicerat den mars 4, 2012 av mattelararen

Katedralen i Siena

That one rotation equals 360 degrees is just a convention, There is nothing partcular about 360 except that it can be divided by many numbers.

Another, and a more fruitful, approach to measuring angles is to use the length of the arc cut out by the angle on the perimeter of the unit-circle.

Since the circumference of the unit-circle is 2π this corresponds to 360 degree

π RADIANS then equal 180degrees and hence 1 radian = 180/π degrees.

It follows that one revolution corresponds to 2π such radians.

http://en.wikipedia.org/wiki/Radian

This reasoning can be expanded to  arbitrary circles having radius r.

Their circumference being 2πr which means 2π radians proving that the angle is equal to 2π radians in this case as well.

To see an attempt to square the circle click on the figure below. Courtesy of Wiking Björkman (my grandfather):

To listen to an 1981 interview (Radio Kronoberg) with him click below
11 Spår 11_(new).mp3

Publicerat i Geometri, Gymnasiematematik(high school math) | Lämna en kommentar