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Calculus: Integrals & Applications

This document discusses multiple integrals. It explains that a double integral generalizes the definition of a single variable integral to two dimensions, and a triple integral generalizes it to three dimensions. It describes how to evaluate double integrals by splitting the region into subregions and integrating over each separately. It also discusses changing variables and order of integration in double integrals, and how to evaluate triple integrals by repeated integration.

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Sai Prajit
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91 views9 pages

Calculus: Integrals & Applications

This document discusses multiple integrals. It explains that a double integral generalizes the definition of a single variable integral to two dimensions, and a triple integral generalizes it to three dimensions. It describes how to evaluate double integrals by splitting the region into subregions and integrating over each separately. It also discusses changing variables and order of integration in double integrals, and how to evaluate triple integrals by repeated integration.

Uploaded by

Sai Prajit
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as DOC, PDF, TXT or read online on Scribd
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UNIT-I

DIFFERENTIAL EQUATIONS OF
FIRST ORDER
AND THEIR APPLICATION
MULTIPLE INTEGRALS

 Let y=f(x) be a function of one


variable defined and bounded on [a,b]. Let
[a,b] be
divided into n subintervals by points x 0 ,… ,x
n such that a=x0 ,………,xn=b. The
generalization of this definition to two
dimensions is called a double integral and to
three dimensions is called a triple integral.
DOUBLE INTEGRALS
 Double integrals over a region R may be
evaluated by two successive integrations.
Suppose the region R cannot be represented by
those inequalities, and the region R can be
subdivided into finitely many portions which
have that property, we may integrate f(x,y)
over each portion separately and add the
results. This will give the value of the double
integral.
CHANGE OF VARIABLES IN DOUBLE INTEGRAL

 Sometimes the evaluation of a double or


triple integral with its present form may not be
simple to evaluate. By choice of an appropriate
coordinate system, a given integral can be
transformed into a simpler integral involving
the new variables. In this case we assume that
x=r , y=r sin and dxdy=rdrd
cos
CHANGE OF ORDER OF INTEGRATION
 Here change of order of integration implies that the
change of limits of integration. If the region of
integration consists of a vertical strip and slide along
x-axis then in the changed order a horizontal strip
and slide along y-axis then in the changed order a
horizontal strip and slide along y-axis are to be
considered and vice-versa. Sometimes we may have
to split the region of integration and express the
given integral as sum of the integrals over these sub-
regions. Sometimes as commented above, the
evaluation gets simplified due to the change of order
of integration. Always it is better to draw a rough
sketch of region of integration
TRIPLE INTEGRALS

The triple integral is evaluated as the repeated
integral where the limits of z are z1 , z2 which
are either constants or functions of x and y;
the y limits y1 , y2 are either constants or
functions of x; the x limits x1 , x2 are constants.
First f(x,y,z) is integrated w.r.t. z between z
limits keeping x and y are fixed. The resulting
expression is integrated w.r.t. y between y
limits keeping x constant. The result is
finally integrated w.r.t. x from x1 to x2 .

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