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Second Fundamental Theorem of Calculus

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					   Second Fundamental Theorem of Calculus
Second Fundamental Theorem of Calculus

We all encounter many type of the problems in the calculus while studying the math but most
of us are generally not familiar with the fundamental theorem of calculus so in this article we
will discuss about the statement of the fundamental theorem of calculus, we will try to explain
the fundamental theorem of calculus and also we will discuss some of the applications of the
Fundamental Theorem of Calculus.

So let us start with the statement of the fundamental theorem of calculus. The fundamental
theorem of calculus is a theorem which establishes a relationship between the idea of the
derivatives and the idea of the integrals.

Before discussing any further details about the fundamental theorem of calculus it should be
noticed that the fundamental theorem of calculus is divided into the two parts which we will
explain in the next two paragraphs.

The first section of the fundamental theorem of calculus describes that any integration which
is of indefinite type can be traced back or we can say that we can reverse it back with the help
of the differentiation.


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This part or the section of the fundamental theorem of calculus is generally known as the first
fundamental theorem of calculus.

The first fundamental theorem of calculus is generally very crucial as it gives the assurance of
the presence of the anti derivatives for the functions which are continuous.

The second section of the fundamental theorem of calculus helps us to perform the definite
integration of a function and thus calculate the integral of the function by utilizing any one of
the infinite existing anti derivatives.

This portion or this section of the fundamental theorem of calculus is generally known as the
second fundamental theorem of calculus.

The second fundamental theorem of calculus has very wide range of the practical utilities as it
very easily simplifies the calculations of the definite integration of the functions.

Now we have seen the different statements of the fundamental theorem of calculus in the
previous paragraphs so let us now understand the physical meaning of the fundamental
theorem of calculus.

Physically the statement of the fundamental theorem of calculus is being given as follows. The
fundamental theorem of calculus states that the addition of the infinitesimally small variations
which occur in a quantity after some time or we can say that over another quantity, brings up a
net variation in that quantity.

Let us consider an example now to have an understanding of the fundamental theorem of
calculus in a more precise way.

Let us suppose that any particle is moving on a straight line then the position x of the particle
is being represented by the notation x ( t ) in which t is representing the time whereas by x ( t )
we mean that x is any function of the time t.

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now if we talk about the derivative of this function then it comes out as the infinitesimally small
variation in the quantity which is dx per the infinitesimally small variation in the time which is
dt. It should be kept in mind here about the fact that the derivative which comes out does not
depend on the time.

In Integral Calculus, there are two different fundamental theorems as follows:

<<-- First Fundamental Theorem of Calculus

<<-- Second Fundamental Theorem of Calculus

In this page, we will learn about these theorems in detail.
From the above two theorem, we infer the following

<<-- The fundamental theorem of integral calculus shows a close relationship between
differentiation and integration.

<<--These theorems give an alternate method evaluating definite integral, without calculating
the limit of a sum.




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