multiple integral

1. DISCOVER . LEARN . EMPOWER
Multiple Integral
INSTITUTE : University Institute of Engineering
Bachelor of Engineering
Subject Name and Code : Mathematics-II
22SMT-125
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Objectives
The Course aims to:
To impart analytical ability in solving mathematical problems as
applied to the respective branches of Engineering
1. To Understand basic concept of Multiple Integral.
2. To learn about Change the order of integration , Change of
Variable and their Application.
Pre-Requisites
1. Basic Knowledge of Integration
2. Basic Knowledge of Solid Geometry

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Topics to be covered in this lecture
1. Multiple Integration
2. Kinds of Multiple Integral
3. Double Integral (in Cartesian & Polar Coordinates)
4. Change of Order
5. Triple Integration
6. Change of Variable
7. Application of Multiple Integration
7.1 Area by Double Integration
7.2 Volume by Double as well as Triple Integration

4. Double Integral
Definition: let f(x,y) be a function in three
dimension space whose domain is x-y plane.
Also let R be any region in x-y plane. Suppose
we divide region R into small sub regions let
be area of sub region then
Double Integral of f(x,y) over region R is
defined as
{∴ Provided Limit Exist}
Where be any point in sub region.
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5. Remarks
1. Represent volume below the surface f(x,y) and above
the region R.
2. If f(x,y) = 1 then Represent Area of region R.
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6. Evaluation of Double Integral in Cartesian Coordinates
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Let Region R is bounded by So we have three cases .
Case 1: when are Constants
Region
Region R is bounded by curves
Then

7. Case-2. When are functions of y & are constants
Region
Region R is bounded by curves
Then
Here limit of x are function of y, we integrate
firstly w.r.t ‘x’ then w.r.t ‘y’
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8. Case-3. When are functions of x & are constants
Region
Region R is bounded by curves
Then
Here limit of y are function of x, we integrate
firstly w.r.t ‘y’ then w.r.t ‘x’
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9. Remarks
1. If variable limit Present then integrate firstly w.r.t. that variable
whose limit are in variable then w.r.t. the variable having constant
limit.
2. When all the limits are constant then integrate according to order of
integration (dxdy or dydx order from left to right)
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10. Evaluation of Double Integral in Polar Coordinates
Here We Evaluate over the
region bounded by and
the curve
(treated as Constant)
Firstly integrate w.r.t. r then w.r.t.
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11. Triple Integral
The triple integral is similar to double integral as a limit of a Riemann
sum.
Definition of the Triple Integral
• Let f(x,y,z) be a continuous function of three variables defined over a
solid ‘S’. Then the triple integral over ‘S’ is defined as
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12. Evaluating Triple Integrals
Case-1. All the limits of integration are constant
Let f(x,y,z) be a continuous function over a solid S ,
, Where “ a , b , c , d , e , f ” all are
constants
Then
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13. Evaluating Triple Integrals
Case-2. If the Limit of Integration are Variable
Let f(x,y,z) be a continuous function over a solid S ,
Then
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Reference Books
H.K. Dass., Higher Engineering Mathematics, S Chand Publishers, 3rd revised
edition, 2014.
N.P. Bali and Manish Goyal, A text book of Engineering Mathematics, Laxmi
Publications, Reprint, 2008.
Erwin Kreyszig, Advanced Engineering Mathematics, 9th Edition, John Wiley &
Sons, 2006.
Online Video Sites:
1. NPTEL (https://nptel.ac.in/courses/111/105/111105122/)
2. Coursera (https://www.coursera.org/lecture/vector-calculus-
engineers/double-and-triple-integrals-lecture-22-6Ifj2)
References

15. THANK YOU
For queries
Email: khushboo.maths@cumail.in
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