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).
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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.

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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.
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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.
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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.

Greatest Integer Function

In the previous post we have discussed about Define Sector of a Circle and In today's session we are going to discuss about Greatest Integer Function.Function are widely used in mathematics. Greatest integer function is one of them, it is also known as the names of floor function or step functions. Here in this blog we are going to discuss the greatest integer function. The greatest integer function for a real number x can be denoted as [x] or [_x_]. This function returns the greatest integer that is less than or equal to x. To be more precise, it rounds downward the rates into closest integer.
[4.9] = 4 (greatest integer less than and equal to 4.9)
[2.99] = 2 (greatest integer less than and equal to 2.99)
[6] = 6 (6 itself as an integer)
[-4.87] = -5 (greatest integer less than -4.87 is -5)
You should remember that all of the above examples are based on a rule that is a greatest integer will be less than or equal to x.
To understand greatest integer function we can plot function on the graph paper.
Example 1: - Determine the greatest integer function for the given numbers.
1.1, 7, -4.8, 2 and -3.6
Solution: -
We have 1.1, 7, -4.8, 2 and -3.6 real numbers.
[1.1] = closest greatest integer value is less than 1.1 is 1.
[1.1] = 1
[7] = 7 is itself an integer that is equal to x so
[7] = 7
[-4.8] = in this closest greatest integer is -5 which is less than -4.8
[-4.8] = -5
[2] = 2 is itself an integer that is equal to x so
[2] = 2
[-3.6] = in this closest greatest integer is -4 which is less than -3.6
[-3.6] = -4

The greatest integer function is defined piecewise.

The area of a circle formula is ∏r ². It helps us to find the area of circle. ICSE class 10 books covers  all the  syllabus in a very  organized and  effective  manner

Define Sector of a Circle

In geometry, Sector of a circle is defined as the area which is covered by two radii and an arc of a circle where arc is the smaller part of the circumference of the circle. It is also known as circular sector or circle sector. Basically, there are two parts of a circle, the first one is sector and the one is segment. Sector of a circle is mainly divided into two parts i.e. quadrant and a semicircle. Quadrant is defined as the quarter of a circle and semicircle is defined as half of a circle. In other words you can say that a sector having the angle 180® is also called as semicircle.


Area of sector of a circle is given by :
K = r² . α/2
Where r is the radius of sector of a circle and α is the central angle.
Perimeter of sector of a circle is given by:
T = C + 2r
  =  αr + 2r
  = r (α + 2)
Where c is the arc length of sector of a circle.
Properties of sector of a circle:
Radius: A radius of a circle is defined as the half of the diameter. It is the maximum distance between the centre of the circle and the circumference and in order to determine the radius we can use the formula of circumference and i.e. [2 x pie x r]
Arc length: Arc is defined as the segment of the circumference and arc length is define as the curve distance between the central angle which is define by the radius.
Central angle: Angle Obtained from the sector to the centre of the circle is known as central angle.
At last sector of a circle and Subtraction Worksheets are also discussed in CBSE Board Previous Year Question Papers for Class 11 and in the next session we willl discuss about Greatest Integer Function.