Higher Engineering Mathematics, Sixth Edition

662 Higher Engineering Mathematics

The circle:

With reference to Fig. FA3.

Area=πr^2 Circumference= 2 πr

πradians= 180 ◦

r s

r



Figure FA3

For sector of circle:

s=rθ (θin rad)

shaded area=^12 r^2 θ (θin rad)

Equation of a circle, centre at (a, b), radius r:

(x−a)^2 +(y−b)^2 =r^2

Linear and angular velocity:

Ifv=linear velocity (m/s),s=displacement (m),

t=time (s),n=speed of revolution (rev/s),

θ=angle (rad),ω=angular velocity (rad/s),

r=radius of circle (m) then:

v=

s

t

ω=

θ

t

= 2 πn v=ωr

centripetal force=

mv^2

r

wherem=mass of rotating object.

Graphs

Equations of functions:

Equation of a straight line: y=mx+c

Equation of a parabola: y=ax^2 +bx+c

Circle, centre (a, b), radius r:

(x−a)^2 +(y−b)^2 =r^2

Equation of an ellipse, centre at origin, semi-axes a

and b:

x^2

a^2

+

y^2

b^2

= 1

Equation of a hyperbola:

x^2

a^2

−

y^2

b^2

= 1

Equation of a rectangular hyperbola: xy=c^2

Irregular areas:

Trapezoidal rule

Area≈

(

width of

interval

)[

1

2

(

first+last

ordinates

)

+

(

sum of remaining

ordinates

)]

Mid-ordinate rule

Area≈

(

width of

interval

)(

sum of

mid-ordinates

)

Simpson’s rule

Area≈

1

3

(

width of

interval

)[(

first+last

ordinate

)

+ 4

(

sum of even

ordinates

)

+ 2

(

sum of remaining

odd ordinates

)]

Vector Geometry

Ifa=a 1 i+a 2 j+a 3 kandb=b 1 i+b 2 j+b 3 k

a·b=a 1 b 1 +a 2 b 2 +a 3 b 3

|a|=

√

a^21 +a 22 +a 32 cosθ=

a·b

|a||b|

a×b=

∣ ∣ ∣ ∣ ∣ ∣

ijk

a 1 a 2 a 3

b 1 b 2 b 3

∣ ∣ ∣ ∣ ∣ ∣

|a×b|=

√

[(a·a)(b·b)−(a·b)^2 ]