1. MATHEMATICS-I

2. CONTENTS
 Ordinary Differential Equations of First Order and First Degree
 Linear Differential Equations of Second and Higher Order
 Mean Value Theorems
 Functions of Several Variables
 Curvature, Evolutes and Envelopes
 Curve Tracing
 Applications of Integration
 Multiple Integrals
 Series and Sequences
 Vector Differentiation and Vector Operators
 Vector Integration
 Vector Integral Theorems
 Laplace transforms

3. TEXT BOOKS
 A text book of Engineering Mathematics, Vol-I
T.K.V.Iyengar, B.Krishna Gandhi and Others,
S.Chand & Company
 A text book of Engineering Mathematics,
C.Sankaraiah, V.G.S.Book Links
 A text book of Engineering Mathematics, Shahnaz A
Bathul, Right Publishers
 A text book of Engineering Mathematics,
P.Nageshwara Rao, Y.Narasimhulu & N.Prabhakar
Rao, Deepthi Publications

4. REFERENCES
 A text book of Engineering Mathematics,
B.V.Raman, Tata Mc Graw Hill
 Advanced Engineering Mathematics, Irvin
Kreyszig, Wiley India Pvt. Ltd.
 A text Book of Engineering Mathematics,
Thamson Book collection

5. UNIT-V
CHAPTER-I:APPLICATIONS OF
INTEGRATION
CHAPTER-II:MULTIPLE INTEGRALS

6. UNIT HEADER
Name of the Course: B.Tech
Code No:07A1BS02
Year/Branch: I Year
CSE,IT,ECE,EEE,ME,CIVIL,AERO
Unit No: V
No. of slides:17

7. UNIT INDEX
UNIT-V
S. No. Module Lecture PPT Slide No.
No.
1 Introduction, Length, L1-5 8-11
Volume and Surface
area
2 Multiple integrals, L6-10 12-15
Change of order of
integration
3 Triple integration , L11-12 16-19
Change in triple
integration

8. Lecture-1
APPLICATIONS OF INTEGRATION
 Here we study some important applications of
integration like Length of arc, Volume,
Surface area etc.,
 RECTIFICATION: The process of finding the
length of an arc of the curve is called
rectification.
 Length of an arc S=∫[1+(dy/dx)2]1/2

9. Lecture-2
LENGTH OF
CURVE(RECTIFICATION)
 The process of finding the length of an arc of
the curve is called rectification. We can find
length of the curve in Cartesian form, Polar
form and Parametric form.
 Length of curve in cartesian form: S= ∫[1+
(dy/dx)2]1/2
 Length of curve in parametric form:
S=∫√(dx/dθ)2+(dy/dθ)2 dθ

10. Lecture-3
ARC LENGTH
 Polar form:
 If r=f(θ) and θ=a, θ=b then
S=∫√r2+(dr/dθ)2 dθ
 If θ=f(r) and r=r1 , r=r2 then
S=∫√1+r2(dθ/dr)2 dr

11. Lecture-4
VOLUME
 If a plane area R is revolved about a fixed line
L in its plane, a solid is generated. Such a solid
is known as solid of revolution and its volume
is called volume of revolution. The line L
about which the region R is revolved is called
the axis of revolution.Volume of the solid can
be found in 3 different forms Cartesian form,
Polar form and Parametric form.
 Volume of the solid about x-axis= ∫пy2dx

12. Lecture-5
FORMULAE FOR VOLUME
 Cartesian form:
Volume of the solid about x-axis=∫пy2dx
Volume of the solid about y-axis=∫пx2dy
 Volume of the solid about any
axis=∫п(AR)2d(OR)
 Volume bounded by two curves=
∫п(y12-y22)dx

13. Lecture-6
SURFACE AREA
 The surface area of the solid generated by the
revolution about the x-axis of the area
bounded by the curve y=f(x).We can find
revolution about x-axis,y-axis,initial line, pole
and about any axis.
 Example: The Surface area generated by the
circle x2+y2=16 about its diameter is 64π

14. Lecture-7
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,…,xn
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.

15. Lecture-8
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.

16. Lecture-9
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 cosθ, y=r sinθ and dxdy=rdrdθ

17. Lecture-10
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.

18. Lecture-11
TRIPLE INTEGRALS
 The triple integral is evaluated as the repeated
integral where the limits of z are z 1 , 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.

19. Lecture-12
CHANGE OF VARIABLES IN TRIPLE
INTEGRAL
 In problems having symmetry with respect to a point
O, it would be convenient to use spherical
coordinates with this point chosen as origin. Here we
assume that x=r sinθ cosф, y=r sinθ sinф, z=r cosθ
and dxdydz=r2 sinθ drdθdф
 Example: By the method of change of variables in
triple integral the volume of the portion of the sphere
x2+y2+z2=a2 lying inside the cylinder x2+y2=ax is
2a3/9(3π-4)