How to Solve Definite Integral Problems

In the previous post we have discussed about How to Solve Antiderivative  and In this blog we are going to discuss about the topic definite integral which is an important topic of calculus. In mathematics calculus is very useful to define the quantities that are not solved with the help of simple algebraic functions and rules. Integration is the process of finding the value that are in the limits defined in the integration function. It also define in therms of area on the xy plane of the graph where limits defines the axis of the graph it is easily explained by an expression as if there is a function f (n) and it will be integrated in the limits p and q that is written as F = ʃ _p ^q f (n) d n and here function F define the area on the plane XY that is bounded with in the x axis and y axis where x = define the area above the x axis and x = q define the area below the x axis.
Whenever a integration process is done within the closed interval that is come into the category of definite integral that is explain as if there is a function p (s) that is integrated with in the interval a and b is expressed as
ʃ_a ^b p (s) d s = lim _{n→ infinity} ∑ n _u = 1 p (s _u) d s
here ∑ n _u = 1 p (s _u) d s define the sum of the values that are come into the interval a to for the function g (s).
 Harmonic Mean is one of the method for finding the measure of central tendency that is used for calculation of quantitative data. ISEET stands for Indian science engineering Eligibility test and Syllabus of ISEET have all the related topics of physics that are categorized according to the section define in the syllabus.

How to Solve Antiderivative

In this blog we are going to discuss about the topic Antiderivative . In calculus antiderivative is define as the function that is opposite to the derivative function. It define as the function f which gives the derivative F that means f ' = F. It is calculate by doing the opposite functioning of differntiation.

Antiderivative is also define as the integration function that is also known as the indefinite integration. It is defined by a simple example as if we take a function f = sin (a) then the derivative of function f is define as the f' = cos (a) and when we find he antiderivative of f' that is equal to f that means antiderivative of function f' = f that means antiderivative of cos (a) is define as

ʃ cos (a) d a = sin (a) + c.

As we know that differentiation and integration are opposite process and differentiation is another name of anti derivation . It follows all the rules of integration as If there is define the limit of the function Êƒp q f (a) d a = f (q) – f (P) . there are also some properties of antiderivative that are as follows:
ʃ a n d a = a n+1 / (n +1) + c. here c is the constant value .

There is one thing have to know before finding the antiderivative that there is several integrals of a function that are different to each other on the basis of their constant term that is generated at the time of integration.

Topic on Cumulative Frequency defines all the related problems of cumulative frequency define into the calculus in a very simple way. ISEET Chemistry syllabus define all the topics of chemistry that helps the student to secure good marks in the chemistry and In the next session we will discuss about How to Solve Antiderivative. 

Define Laplace Transform

Laplace transform is a type of integral transform that are popularly used in the field of physics and engineering. In mathematical notation the concept of Laplace transform can be represented as L f(a). The above given notation can be consider as a linear operator for a function f (a). Here the value ‘a’ can be considered as a real argument, which should be greater than equal to zero. When the Laplace transform are performed on the above given function then it transform them into f(a) → f(b). Here f(b) can be consider as function with complex argument ‘b’.

Normally the concept of Laplace transform is related to the Fourier transform. But there is big difference between them. Laplace transform resolves a function into moments whereas in same aspect Fourier transform perform the task of representing a function in a series of modes. The concept of Laplace transform is formed by Great mathematician Mr. Pierre Laplace at the time when he performing his experiment on Probability theory. In the below we show you how a Laplace transforms works:

Suppose there F(a) is a function defined as [0, ∞]. Now laplace transform for F(a) can be defiend  as new function :

L(f(b)) = ⌠0∞ e-ba f(a)da = lim α→∞ ⌠0α e ba f(a)da

the basic meaning of above given mathematical notation is that there is a function f(a) on which Laplace transform are performed that gives the new function f(b) with a complex number b. The scope of integral depends on types of function of interest. The basic reason behind the use of Laplace transform is that to reduce a differential equation to an algebra problem. In mathematics, Laplace transform works well when forcing function in differentiation equation gets more complicated. The Statistical Inference is a mathematical process that perform the task of drawing conclusion from data. karnataka education board came into existence into real world in the year of 1966. The main work of karnataka education board is conducting SSLC examinations and In the next session we will discuss about How to Solve Antiderivative.

 

Pie chart

In the previous post we have discussed about How to find Intersection and In today's session we are going to discuss about Pie chart. In statistics we use various kinds of graphs to summarize the numerical data in graphical format. Here we are going to discuss the Pie chart, besides bar graphs, histograms, stem plots, dot graph, box plots, pie chart is also one of the type of graph.
With the help of pie chart we can represent qualitative and quantitative both the type of data. Although it is used to show the categorical type of data. Pie charts are very effective to showing how the specific parts are associated in the whole distribution. In pie charts data are shown in the circular format. It means different types of data are shown in different sectors. For example below is the figure that describes following thing:-

The red color covering more area than purple and pink colors, this is showing population of China which is highest among all countries
After that the purple color covering more area than pink color but less than red color, this is showing population of India which is second highest among all countries.  (know more about Pie chart, here)
And in third one sector the pink color covering less area than red and purple colors, this is because, America is third country that has highest population.
So in above pie chart there are three types of data that is divided according to the population.
Using colors, labels, shades and pictures is a good habit at the time of designing pie chart, by doing this not only us but also everyone can understand the problem and problem solution easily. This approach makes the pie charts attractive. Pie charts are the visual comparisons of the distribution.


Scatter Plots are the type of mathematical diagram that is used to show the values of two variables. Maharashtra secondary board is very dedicated regarding SSC and HSC examinations.

How to find Intersection

Set is the collection of the elements. We can perform much operation on the sets like union, intersection, difference and many more. So here we are going to discuss the intersection operation. Intersection means a set that is made by combining common elements of two or more sets. Intersection symbol is ∩. Suppose we have two sets P and Q then we can represent them as P∩Q. Let define the P and Q sets by an example.
Set P contain elements such as 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,
Set Q contain elements such as 3, 5, 7, 8, 9, 12, 19, 23
So the intersection elements will be 3, 5, 7, 8, 9, it is because 3, 5, 7, 8, 9 elements are same in both of the sets
We can represent the all scenario by making a figure.

The shaded area of figure is showing intersections elements as we discussed above.
Let’s take another example to understand the intersection
Example: - Suppose we have two sets A and B
Set A = 2, 4, 5, 7, 8, 9, 12, 16, 19, 21, 23
Set B = 1, 3, 6, 9, 19, 21, 23, 29, 30
Than the intersection will be represented as A∩B = 9, 19, 21, 23
We can find the intersection for more than two sets just we have to take those numbers that are common in all three sets. Suppose we have A, B and C three sets than they can be represented as A∩B∩C.
At the time of dealing with central tendency one question is arises in everyone mind that how to find the mean, mean is the average of given distribution. Tamilnadu state board syllabus is very strong so due to this it has first rank among all boards.

How to find Intersection

Set is the collection of the elements. We can perform much operation on the sets like union, intersection, difference and many more. So here we are going to discuss the intersection operation. Intersection means a set that is made by combining common elements of two or more sets. Intersection symbol is ∩. Suppose we have two sets P and Q then we can represent them as P∩Q. Let define the P and Q sets by an example.
Set P contain elements such as 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,
Set Q contain elements such as 3, 5, 7, 8, 9, 12, 19, 23
So the intersection elements will be 3, 5, 7, 8, 9, it is because 3, 5, 7, 8, 9 elements are same in both of the sets
We can represent the all scenario by making a figure.

The shaded area of figure is showing intersections elements as we discussed above.
Let’s take another example to understand the intersection
Example: - Suppose we have two sets A and B
Set A = 2, 4, 5, 7, 8, 9, 12, 16, 19, 21, 23
Set B = 1, 3, 6, 9, 19, 21, 23, 29, 30
Than the intersection will be represented as A∩B = 9, 19, 21, 23
We can find the intersection for more than two sets just we have to take those numbers that are common in all three sets. Suppose we have A, B and C three sets than they can be represented as A∩B∩C.
At the time of dealing with central tendency one question is arises in everyone mind that how to find the mean, mean is the average of given distribution. Tamilnadu state board syllabus is very strong so due to this it has first rank among all boards.

How to tackle Logarithm Problem

In the previous post we have discussed about Greatest Integer Function and In today's session we are going to discuss about How to tackle Logarithm Problem. The definition of the logarithm of a number x with respect to the base b is nothing but the exponent to which the base b has to be raised to get the number x. In other words, the logarithm of x to base b is the solution y of the equation.
b^y = x
The logarithm is denoted by log_b. The conditions for a logarithm to be defined are: the base b must be a positive real number and must not equal to 1 and x must also be a positive number.
So we can see that the whole idea of logarithm is just to reverse the operation of exponentiation, which is nothing but raising a number to a power. For example, the third power or you can say cube of 3 is 27, because 27 is the product of three factors of 3:
b^y = x
It says that the logarithm of 27 with respect to the base 3 is 3.
If I tell you in a simple way then logarithm of any number is the exponent by which the base of the logarithm has to be raised to produce that number. For ex., the log of 100 to the base 10( which is the common logarithm) is 2, because 100 is 10 to the power 2:100 = 10² = 10 × 10.
 So, if x = b^y, and we take log both side,
 Then only y will remain on R.H.S because log is the reverse of exponentiation and b will go to the L.H.S as base of log(x) and written as y = log_b(x), so log₁₀(100) = 2.

The common logarithm, which is nothing but log with respect to base 10, has many applications in science and engineering.
We can reduce wide range quantities to smaller level by using logarithmic scales.
In order to get help in understanding the topics more: logarithm, average rate of change, you can just visit our next article. All this topics are well illustrated in CBSE Text Books.