662 Higher Engineering Mathematics
The circle:
With reference to Fig. FA3.
Area = πr
2 Circumference = 2πr
π radians = 180
◦
s
r
r
␪
Figure FA3
For sector of circle:
s = rθ (θ in rad)
shaded area =
1
2 r 2 θ (θ in rad)
Equation of a circle, centre at (a, b), radius r:
(x − a)
2
+ (y − b)
2
= r
2
Linear and angular velocity:
If v = 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
= 2πn v = ωr
centripetal force =
mv 2
r
where m = 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: x y = 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
If a = a 1 i + a 2 j+ a 3 k and b = 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 2
1 + a 2
2 + a 2
3 cos θ =
a · b
|a| |b|
a × b =
i
j k
a 1 a 2 a 3
b 1 b 2 b 3
|a × b | =
[(a · a)(b · b) − (a · b) 2 ]