Advice 1: How to calculate definite integral in Excel

Let's see how to calculate definite integral table of the given function using the Excel program from Microsoft Office.
How to calculate definite integral in Excel
You will need
  • - computer with installed application MS Excel;
  • table specified function.
Let's say we have a table given a certain value. For example, let it be the accumulated dose of radiation during air travel. Let's say there was such an experiment: a man with a dosimeter was on a plane from point A to point B, and periodically measured dosimeter dose rate (measured in microsieverts per hour). You might is surprised, but in normal flight, the person receives a radiation dose 10 times higher than the background level. But the effect is transient and therefore not dangerous. Based on measurements we have a table like this format: Time - dose.
Tabular specification
The essence of the method is that the definite integral is the area under the graph of the desired size. In our example, if the flight lasted almost 2 hours, from 17:30 to 19:27 (see the figure), to find the accumulated dose, it is necessary to determine the area of the shape under the graph of dose - schedule table predetermined value.
Definite integral - the area under the figure
To calculate the integral we will be the most simple but quite accurate method - the method of trapezoids. Let me remind you, every curve can be divided into a trapezoid. The sum of the areas of these trapezoids and is the required integral.
Area of a trapezoid is defined simply: the sum of the bases multiplied by the height. The reason in our case is a tabular measured values of dose rate for 2 consecutive time interval, and the height is the difference of time between two measurements.
Calculating area of a trapezoid
In our example of measuring the dose of radiation is given in µsv/h. Translate this in µsv/min, since the data are given with a frequency of 1 time per minute. It is necessary to harmonize units of measure. We can't take the integral over time measured in minutes from the value measured in hours.
For translation, simply divide the dose rate in µsv/h by the line 60. Add another column to our table. The illustration in column "D" in line 2, enter "=C2/60". And then using the fill handle (the mouse pull the black rectangle in the lower right corner of the cell) extend this formula to all cells in column "D".
Conversion of units
Now we need to find area of trapezoids each time. In column "E" will be calculated by the above formula of the area of the trapezoids.
The sum of bases is half the sum of the two consecutive dose rate from column "D". As these go with a period of 1 time per minute, and we take the integral time, expressed in minutes, the height of each trapezoid is equal to one (time difference between each two consecutive measurements, for example, 17ч31м - 17ч30м = 0ч1м).
Get the formula in cell E3: "=1/2*(D2+D3)*1". It is clear that "*1" can not write, I did it just for the sake of completeness. The picture explains everything more clearly.
Similarly, using the fill handle distributed by formula to the whole column. Now in each cell of column "E" is calculated, the accumulated dose over 1 minute of flight.
Calculating area of trapezoids
It remains to find the sum of the calculated squares, and trapezoids. In cell "F2" to write the formula "=SUM(E:E)", this is the desired integral is the sum of all values in column "E".
You can do a little bit more difficult to determine the accumulated dose at different points of the flight. To do this, in cell "F4" will enter the formula "=SUM(E$3:E4)" and fill handle will extend the entire column "F". The symbol "E$3" tells Excel what to change the index of the first cell, from which a score, it is not necessary.
We construct a graph in columns "F" and "A", i.e. the change in the accumulated dose of radiation in time. It is clearly seen the increase of the integral, as it should be, and the final value is accumulated over a two-hour flight radiation dose equal to approximately 4.5 microsievert.
Thus, we just found the definite integral of the given function table in Excel to a real physical example.
Evaluation of the integral

Advice 2: How to calculate the integral of the function

Integral calculus is the part of mathematical analysis, basic concepts which primitive function and integral, its properties and calculation methods. The geometric meaning of these calculations – finding the area of curvilinear trapezoid bounded by the limits of integration.
How to calculate the integral of the function
As a rule, the evaluation of the integral boils down to, to bring the integrand to the tabular view. There are many table-valued integrals, which facilitates the solution of such problems.
There are several ways to bring the integral to a convenient form: direct integration, integration by parts, the substitution method, the introduction under the sign of the differential, the Weierstrass substitution, etc.
The method of direct integration is the successive reduction of the integral to the tabular view by using elementary transformations:∫соѕ2 (x/2)DX = 1/2•∫(1 + cos x)DX = 1/2•∫DX + 1/2•∫cos xdх = 1/2•(x + sin x) + C, where C is a constant.
Integral has many possible values based on the properties of the integral, namely the presence of summable constants. Thus, it was found in the sample solution is shared. The private solution of the integral is called the total at a certain constant value, for example, With=0.
Integration by parts is used when the integrand is a product of algebraic and transcendental functions. The method formula:∫udv = u•v - ∫vdu.
Because the position of the multipliers in the product do not matter, as the function u to choose one part of the expression, which after differentiation is simplified. Example:∫x·ln xdx = [u=ln x; v=x; dv=xdx] = x2/2·ln x – ∫x2/2·dx/x = x2/2·ln x – x2/4 + C.
The introduction of the new variable is a technique for method lookup. This changes and she integrand, and its argument:∫x·√(x - 2)dx = [t=x-2 → x = t2+2 → dx=2·tdt] = ∫(t2 + 2)·t·2·tdt = ∫(2·t^4 + 4·t2)dt = 2·t^5/5 + 4·t3/3 + C = [x=t2+2] = 2/5·(x - 2)^(5/2) + 4/3·(x - 2)^(3/2) + C.
Method of introducing, under the sign of the differential involves the transition to new functions. Let ∫f(x) = F(x) + C and u = g(x), then ∫f(u)du = F(u) + C [g’(x) = dg(x)]. Example:∫(2·x + 3)2dx = [dx = 1/2·d(2·x + 3)] = 1/2·∫(2·x + 3)2d(2·x + 3) = 1/6·(2·x + 3)3 + C.

Advice 3: What is measured by the radiation

Radiation is ionizing radiation, which is divided into several types. High doses of radiation are dangerous to health and life. To measure the effects of radiation on the body is used the unit sievert. More common size measurement of radiation – grey – means the radiation dose absorbed by the substance.
What is measured by the radiation

What is radiation?

Invisible and invisible radiation can kill a person within hours or days. Ionizing radiation is naturally found throughout the surface of the Earth, but in too small quantities. But there are places where background radiation is much higher, and when accidents at nuclear power plants during the nuclear bombing and in other situations, the radiation dose may exceed the norm by several times.

From a scientific point of view, the radiation is a stream of microscopic particles that can ionize meeting on their way substance. Such influence in living cells of biological organisms, including human, are formed by the alien, not its inherent chemical compounds. The correct course of intracellular processes stops, the cell structures are destroyed, they gradually die.
If the dose is small, the cells can recover from such damage.

Measurement of radiation

There are several units for measuring radiation, which are used depending on the situation. If the measured absorbed dose, i.e. that dose of radiation which is absorbed by a particular mass unit, it uses so-called grey, which actually represents the number of joules per kilogram.
This unit is named in honor of one of the most prominent figures among the scientists involved in radiobiology – Lewis gray.

But this measurement is not used when describing the effects of radiation on the human body. To do this, use a different value, which measures effective dose. It is called the sievert, the unit used only since 1979, but all modern dosimeters that determine the radiation show results in this unit of measurement is named after the physicist – Rolf Sievert.

Effective dose depends on several parameters: the type of radiation (there are alpha-, beta - and gamma-rays) from the radiation direction (different human organs in different ways resist radiation). In certain conditions, it turns out the coefficient of biological hazards, which is multiplied by the number of gray, that is, absorbed dose, and get the value in sieverts.

Such a famous unit of measure of radiation like x-rays, applies only to gamma radiation, or x-ray. One sievert is equal to approx one hundred x-rays.

To determine the activity of a radioactive source, i.e. the number of decays of nuclei for a certain period of time, apply another unit is the Becquerel. Kinetic energy of particles is measured in electronvolt.
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