One of the earliest mathematicians who made wonderful discoveries to compute the areas and volumes of geometrical objects was Archimedes.

**Applications
of Integration**

*“Give me a place to stand and I will move the earth” - ***Archimedes**

** **

One of
the earliest mathematicians who made wonderful discoveries to compute the areas
and volumes of geometrical objects was Archimedes. Archimedes proved that the
area enclosed by a parabola and a straight line is 4/3 times the area of an
inscribed triangle (see^{ }Fig. 9.1).

He
obtained the area by segmenting it into infinitely many elementary areas and
then finding their sum. This limiting concept is inbuilt in the definition of
definite integral which we are going to develop here and apply the same in
finding areas and volumes of certain geometrical shapes.

** **

Upon
completion of this Chapter, students will be able to

• define
a definite integral as the limit of a sum

• demonstrate
a definite integral geometrically

• use
the fundamental theorem of integral calculus

• evaluate
definite integrals by evaluating anti-derivatives

• establish
some properties of definite integrals

• identify
improper integrals and use the gamma integral

• derive
reduction formulae

• apply
definite integral to evaluate area of a plane region

• apply
definite integral to evaluate the volume of a solid of revolution

We
briefly recall what we have already studied about anti-derivative of a given
function *f* ( *x*) . If a function *F* ( *x*) can be found such that *d/dx* *F*
( *x*) = *f*
( *x*) , then the function *F* ( *x*)
is called an **anti-derivative
**of** ***f*** **(** ***x*)** **.

It is
not unique, because, for any arbitrary constant *C* , we get *d/dx* [*F* ( *x*)
+
*C*] = *d/dx*
[*F* ( *x*)] = *f*
( *x*) .

That is,
if *F* ( *x*) is an anti-derivative of *f*
( *x*) , then the function *F* ( *x*)
+
*C* is also an anti-derivative of the
same function *f* ( *x*) . Note that all anti-derivatives of *f* ( *x*)
differ by a constant only. The anti-derivative of *f* ( *x*) is usually called
the **indefinite
integral** of *f* ( *x*) with respect to *x* and is denoted by ∫ *f*
( *x*)*dx* .

A
well-known property of indefinite integral is its **linear property** :

∫ [*α*
*f* ( *x*) + *β* *g* ( *x*)]*dx* = *α* ∫
*f* ( *x*)*dx* + *β* ∫
*g* ( *x*)*dx* ,
where *α* and *β* are constants.

We list
below some functions and their anti-derivatives (indefinite integrals):

Tags : Mathematics , 12th Maths : UNIT 9 : Applications of Integration

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