Friends in today’s session I am going to lay stress on a very interesting and a bit complex
topic of mathematics that is Definite Integrals. This subject is about the basic concept of
A Definite Integral of a function is basically the signed area of a given region which is covered
by its graph. Integration is a very important topic of calculus.
It is a reverse process of differentiation or we can say it is anti differentiation of a function.
Integrals are used in integration.
Suppose we have a function say f of any variable say y with a given interval [p, q] then its
definite integral can be represented as:
This is defined as the full signed area of the region in yx plane which is covered by its own
graph of the function f.
Integral also represents the antiderivative of a function say F. The derivative of this function F
is the given function f. Now the function F is known as the indefinite integral and can be
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The principle of the integration was first developed and formulated by Sir Isaac Newton and
his friend Gottfried.
By using the fundamental theorem of calculus The integration is related to the differentiation
as if a function say f is a continuous and real valued function that is defined on a closed
interval of [p, q] the anti derivative F of function f will be known as the definite integral of the
function f over that given interval and it can be represented as:
= F(q) – F(p)
The integration and differentiation are the basic roots of the calculus. These both have various
applications in physics, engineering etc.
A line integral Is basically formulated for the functions which consist of two or more variables
whose interval of integration was replaced by a certain curve which connects or joins two
points on the plane.
A surface integral is the same as the line integral except the curve. The curve is replaced by
the surface which is in the 3 dimensional spaces.
The function which has an integral is called Integral. The function for which we calculate an
integral is known as the integrand.
And the region or space over which we integrate the function is known as the Domain of The
Integration. Basically this domain is an interval in which we give the lower limit and upper limit
of the interval, which are said to be limits of integration.
If the domain or the region is undefined for any given function then it is always considered as
The function for which we calculate the integral or the integrand can be a function consisting
of one or more variables.
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The domain of the integration can be anything like Area, Volume, A region, or even a space
with no geometrical structure.
Some of the fundamental concepts of definite integrals can be used as definite integral
solvers. Let us study the fundamental concepts one by one.
1) A definite integral of a function f(x) in an interval is the area under the graph of the function
between the given limits. This gives rise to the concept that a definite integral is the limit of a
2) As already mentioned ∫baf(x)dx = F(b) – F(a), where F(x) is the anti derivative of the given
integral. As a corollary, ∫baf(x)dx = 0.
3) If the limits of a definite interval are changed then the value of the given integral becomes
negative. That is, ∫baf(x)dx = - ∫abf(x)dx.
4) If the given interval of a definite integral is expressed as sum of two sub intervals then the
definite integral can be be expressed as a sum of two definite integrals with the limits now
representing the sub intervals. That is, if c is an intermediate point in the given interval [a, b],
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