University Institute of Engineering

DEPARTMENT – AU1- AU5

Bachelor of Engineering
Subject Name: Mathematics-II
Subject Code: 23SMT-125

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 COURSE OUTCOMES
CO Title Level
Will be
Number covered in
Remember the concepts based upon partial differentiation, multiple Remember this
CO1 integrals and vector calculus. lecture

CO2 Understand the concepts of the partial differentiation, multiple Understand

integrals and vector calculus.

Apply concepts of partial differentiation, multiple integrals and Apply

CO3 vector calculus and to use in practical problems.

CO4 Analyze concepts of partial differentiation, multiple integrals and Analyze

vector calculus and its application of analysis to Engineering
problems.

Create the solution for real problems based upon partial Applications
CO5 differentiation, multiple integrals and vector calculus.

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Double Integrals

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 Triple Integrals

Again, the triple integral always exists if f is continuous. We can choose the
sample point to be any point in the
sub-box, but if we choose it to be the point (xi, yj, zk) we get a simpler-looking
expression for the triple integral:

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 Triple Integrals
Just as for double integrals, the practical method for evaluating triple
integrals is to express them as iterated integrals as follows.

The iterated integral on the right side of Fubini’s Theorem means that we
integrate first with respect to x (keeping
y and z fixed), then we integrate with respect to y (keeping
z fixed), and finally we integrate with respect to z.
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 Triple Integrals
There are five other possible orders in which we can integrate, all of
which give the same value.

For instance, if we integrate with respect to y, then z, and then x, we have

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 Example 1
Evaluate the triple integral B xyz2 dV, where B is the rectangular box
given by

B = {(x, y, z) | 0  x  1, –1  y  2, 0  z  3}

Solution:
We could use any of the six possible orders of integration.

If we choose to integrate with respect to x, then y, and then z, we obtain

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Example 1 – Solution cont’d

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 Triple Integrals in Cartesian Coordinates
 The integral of a function f(x,y,z) over a 3D object D, is given by

 D
f ( x, y, z)dV  D f ( x, y, z)dxdydz
dV = dxdydz represents an element
of volume

 The limits on the

integration depend on the
shape of the body D
1
5
 Triple integrals: limits of integration
 Assuming we integrate with respect to z, then y,
then x, the innermost limits may depend on the
other two variables (x and y), the middle limits
may depend on the outer variable (z), whereas the
outer limits are constants.
 The main task is to determine the correct
limits on x, y, z:

 D
u( x, y, z)dzdydx
xb
⎡ y  g 2( x )
⎛⎜ z f 2( x, y )
⎞ ⎤
    u( x, y, z)dz dy⎟ dx⎥
xa ⎢⎣ y  g 1 ( x ) ⎝ z  f1 ( x, y )
 For engineering applications shapes that are important
include: box, cylinder, cone, tetrahedron, sphere.
most ⎦ 3
 Engineering Application of Triple Integrals I
 Volume V of a region D:
V  D dxdydz
 Mass for a body D with density (x,y,z):
m  D  ( x, y, z)dxdydz

 Center of mass for a body D with density (x,y,z)

~x    x ( x, y, z)dxdydz ,
D

m
~y   D y ( x, y, z)dxdydz , 1
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 Engineering Application of Triple Integrals II
 Moment of inertia about the x-axis (Ix) and the y-axis (Iy):
Ix   D   ( x, y, z)dxdydz;
2 2
( y z )
Iy     I z  ..
2 2
D
( x z ) ( x, y, z)dxdydz,

 Total charge for a body with charge density (x,y,z)

Q   ( x, y, z)dxdydz;
D

 Total electrostatic energy (W) stored in a region with electrostatic

filed E
+
r2
Wk   D
E dxdydz; -
1
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 Example 1 (P9-15-6): Evaluate the integral:

Example 2 (P9-6.15)
4 3 22 z / 3
Sketch the region D whose volume V is
given by the integral: 
0 0 0
dxdzdy

Example 3 (P9-15.21)
Find the volume bounded by: x=y2,
4- x=y2, z=0 and z=3

Example 4 (P9-15.27)
Find the center of mass of the solid
bounded by: x2+z2=4,, y=0 and y=3 if the
density  = k y Give details of solutions 6
 Cylindrical Coordinates
Cylindrical coordinates are good for describing solids that are
symmetric around an axis.
Naming convention: a point P(x,y,z)  (r, , z)
• r varies from 0 to ;  varies from 0 to 2, z varies from -  to

•Relation to Cartesian coordinates (Switching) :
•x  r cos , y  r sin ; z  z;

r  x 2 y 2 ,
  tan-1 ( y / x)
 Cylindrical Coordinates

Differential
r length
d:  dr rˆ r d ˆ  dz zˆ
L
Differential surface(dS  dS nˆ) :

r
r  cons tan t, d s  r d dzaˆ r

r
  cons tan t, d s  dr dz aˆ z

r
z  cons tan t, d s  rdr d aˆ
Differential volume : dV  r dr d dz 21
 Triple Integrals in Cylindrical Coordinates

express dV as

 f ( x, y, z)dxdydz  D f (r cos , r sin  , z)rdrddz

st
ir
z-f
D

€
 ⎡ r  g 1( ) ⎛⎜ z f 2( r, ) u(r, , z)dz rdr
⎞ d ⎤ 
   ⎢⎣  r g 1( ) ⎝  z1 f ( r, )
⎟
⎠ €
⎥⎦
ast
 -l 22
 Example 1 (P9-15-45): Convert the following equation to cylindrical
x2 coordinates:
 y2  z2  1

Example 2
Use triple integrals in cylindrical coordinates to find
the volume V bounded by:

x 2  y2  4, x 2  y 2  z 2  16, z  0

Give details of solutions

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 Spherical Coordinates
Spherical coordinates are good for describing
solids that are symmetric around the point.

Naming convention: a point P(x,y,z)  (,, )

 varies from 0 to ;  varies from 0 to 
 varies from 0 to 2,

Relation to Cartesian coordinates (switching):

x   sin cos , y   sin sin , z  

cos
  x 2 y 2  z 2 ,
  tan-1 ( y / x),   tan1 ( x2 y 2
/ z)
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 Triple Integrals in
Spherical Coordinates

express dV as

 f ( x, y, z)dxdydz   D     ddd

2
D
u( , , ) sin
Substitue : x   sin cos , y   sin sin , z   cos
⎡ ⎛ ⎞⎟  2d ⎤sin  d
I    ⎜  u(,, )d ⎥
⎢ ⎝ € ⎠ €
first l a s t
⎣ - - 25
 Example 1: Convert the following equation to spherical coordinates:

z 2  3x 2  3y 2
Example 2 Find the volume bounded by:

x2 y 2
 z 2
 4, y  x, y  3 x, z  0, First Octant

Example 3
Find the moment of inertia about the z axis of the solid:
x y z a
2 2 2 2 The density  = k.
v

Iz    ( x  y )  ( x, y, z)dxdydz; 
2 2
V v Spherical Coordinates

Give details of solutions 14

 Introduce the del operator in both cylindrical and
spherical coordinates through these examples.

Optional Homework:
Given the electrostatic field E  kr3aˆ r
Calculate the charge density v:
r 1  1  E r r
 v  .E  rE r r E  r
z

Calculate the total charge and stored energy in a region bounded by: 0 § r § 1, 0
§  § 2 and 0 § z § 3
Optional Homework:
Given the electrostatic field E  k 2aˆ
Calculate the
r
charge density v:
  . 
v E
1
 2 
 E  sin 
2

1 
sin E sin  
1 E 

Calculate the total charge and stored energy in a region bounded by: 0 §  § 1, 0
§  §  and 0 §  § 2 15
 Triple Integrals
In particular, if the projection D of E onto the xy-plane is a type I plane
region (as in Figure 3).

A type 1 solid region where the projection D is a type I plane region

Figure 3

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 Applications of Triple Integrals
We know that if f(x)  0, then the single integral represents the area under
the curve y = f(x) from a to b, and if f(x, y)  0, then the double integral D
f(x, y) dA represents the volume under the surface z = f(x, y) and above
D.

The corresponding interpretation of a triple integral

 E f(x, y, z) dV, where f(x, y, z)  0, is not very useful because it would
be the “hypervolume” of a
four-dimensional object and, of course, that is very difficult
to visualize. (Remember that E is just the domain of the function f; the
graph of f lies in four-dimensional space.)
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 Applications of Triple Integrals
Nonetheless, the triple integral E f(x, y, z) dV can be interpreted in different
ways in different physical situations, depending on the physical interpretations
of x, y, z and f(x, y, z).

Let’s begin with the special case where f(x, y, z) = 1 for all points in E. Then
the triple integral does represent the volume of E:

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 Applications of Triple Integrals
For example, you can see this in the case of a type 1 region by
putting f(x, y, z) = 1 in Formula 6:

and we know this represents the volume that lies between the surfaces z =
u1(x, y) and z = u2(x, y).

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 Example
Use a triple integral to find the volume of the tetrahedron T
bounded by the planes x + 2y + z = 2, x = 2y, x = 0, and
z = 0.

Solution:
The tetrahedron T and its projection D onto the xy-plane are shown in
Figures 14 and 15.

Figure 14 Figure 15 32
Example – Solution cont’d

The lower boundary of T is the plane z = 0 and the upper boundary is the
plane x + 2y + z = 2, that is, z = 2 – x – 2y.

Therefore we have

(Notice that it is not necessary to use triple integrals to compute volumes.

They simply give an alternative method for setting up the calculation.)

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 Applications of Triple Integrals

•All the applications of double integrals can be immediately extended to

triple integrals.
•For example, if the density function of a solid object that occupies the
region E is  (x, y, z), in units of mass per unit volume, at any given
point
• (x, y, z), then its mass is
and its moments about the three coordinate planes are

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 Applications of Triple Integrals
The center of mass is located at the point where

If the density is constant, the center of mass of the solid is called the
centroid of E.

The moments of inertia about the three coordinate axes are

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 Applications of Triple Integrals
The total electric charge on a solid object occupying a region E and
having charge density  (x, y, z) is

If we have three continuous random variables X, Y, and Z, their joint density

function is a function of three variables such that the probability that (X, Y, Z)
lies in E is

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 Applications of Triple Integrals
In particular,

The joint density function satisfies

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 References

Books
• (i)B. S. Grewal, “Higher Engineering Mathematics”,
Khanna Publication, 2016.(
https://www.scribd.com/document/323240170/Higher-E
ngineering-Mathematics-Dr-B-S-Grewal-Khanna-Publisher
s-pdf
)
• (ii) P. N. Wartikar and J. N. Wartikar, “Applied
Mathematics”, Prentice Hall India, 1999.
Websites
• https://en.wikipedia.org/wiki/Evolute
• http://www.sakshieducation.com/Engg/EnggAcademia/C
ommonSubjects/M1-CurvatureEvolutes&EnvelopesCurveT
racing.pdf
Video Links 38
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