# single_integral

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```					    Unit Four
B.Tech Ist Semester
Content
Arc length
Surface area of solid of revolution.
Volume of solid of revolution.
Arc Length
Length of arc s of the
Parametric curve y=f(x) between the points x=a and x=b is
b
  dy  2 
s=∫          1 +    dx
a     dx  
          
Polar curve r=f(θ) between the points θ=α and θ=β
β
 2  dr  2 
s=∫           r +       dθ
α       
     dθ  
Parametric curve x=f(t), y=f(t) between the ponts t=a and
t=b is              b
 dx   dy  
2       2

s=∫       +    dt
a    dt   dt  
              
e.g.:
(i) Find the length of arc of the parabola x2=4ay
measured from the vertex to one extremity of the
latus –rectum.

[        (
Ans. a 2 + log 1 + 2   )]
(ii) Find the entire length of the cardioid r=a(1+cosθ).

Ans .    2a
Surface area of solid of revolution
(i) About the x-axis: The surface area of the solid
generated by the revolution about x-axis, of the arc
of the curve y=f(x) from x=a to x=b.
b

∫ 2πy ds
a

(ii) About the y-axis: The surface area of the solid
generated by the revolution about x-axis, of the arc
of the curve x=f(y) from y=a to y=b.
b

∫ 2πx ds
a
For polar-coordinates
(iii) About the initial line: The surface area of the solid
generated by the revolution about initial line, of the
arc of the curve r=f(θ) bonded by θ=α and θ=β is
β

∫ 2π r sin θ ds
α
(iv) About the line θ=π/2: The surface area of the solid
generated by the revolution about line θ=π/2, of the
arc of the curve r=f(θ) bonded by θ=α and θ=β is
β

∫ 2π r cosθ ds
α
e.g. :
(i) Find the surface area of the solid generated by the
revolution of the asteroid x=a cos3t , y=a sin3t, about
the y-axis.

12
Ans .    π a2
5

(ii) Find the surface of the solid formed by revolving the
cardioid r=a(l + cosθ) about the initial line.
32π a 2
Ans .
5
Volume of solid of revolution
About the x-axis: The volume of the solid generated
by the revolution about the x-axis, of the area
bounded by the curve y=f(x), the x-axis and the
ordinates x=a, x=b is b
∫ π y dx
2

a
About the y-axis: The volume of the solid generated
by the revolution about the y-axis, of the area
bounded by the curve x=f(y), the y-axis and the
ordinates y=a, y=b is b

∫ π x dy
2

a
Volume of solid of revolution
About any axis: The volume of the solid generated
by the revolution about the any axis LM of area
bounded by the curve AB, the axis LM and the
perpendicular AL,BM on the axis, is
OM                                              B

∫
OL
π ( PN ) 2 d (ON )             A
P

O     L     N      M
e.g.:
(i) Find the volume of the solid formed by the
revolution of the cissoid y3(2a-x)=x3 about its
asymptote.
Ans . 2π 2 a 3

(ii) Find the volume generated by the revolution of the
area bounded by x-axis, the centenary y=c cosh x/c
and the ordinates x=±c, about axis of x.
Ans . π c 3 (1 + sinh 1 cosh 1)
Volume of solid of revolution for polar curve:
The volume of the solid generated by the revolution of
the area bounded by the curve r=f(θ) and the redii
vector θ=α, θ=β
(a) about initial line θ=0 is
β
2 3
∫ 3 π r sin θ dθ
α

(b) about initial line θ=π/2 is
β
2 3
∫ 3 π r cosθ dθ
α
Volume Between two curve
The volume of the solid generated by the revolution
about the x-axis of the area bounded by the curves
y=f(x), y=g(x), and the ordinates x=a, x=b is
b

∫π (y
2       2
1       − y2 ) dx
a

where y1 is the ‘y’ of the upper curve and y2 that of
the lower curve.
e.g.:
(i) Find the volume of the solid formed by revolving a
loop of lemniscate r2=a2 cos 2θ about the initial
line.                     π a3  1             1
Ans .      
4  2
(
log 2 + 1 − 
3
)
(ii) The area bounded by the parabola y2=4x and the
straight line 4x-3y+2=0 is rotated about the y-axis,
obtained the volume of the solid formed by this
revolution.
π
Ans .
20

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