List of formulae AWS
List of formulae Laws of indices:
Areas of plane figures:
a m × a n = a m+n a m/n =
√ n m a
am an
= a m−n (a m )n = a mn
a −n =
a0 = 1 b
Quadratic formula: If ax 2 + bx + c = 0
1 an
Area = l × b
(i) Rectangle
√ −b ± b2 − 4ac x= 2a
then
Equation of a straight line:
l
(ii) Parallelogram
Area = b × h
y = mx + c
Definition of a logarithm: If y = a x
then
h
x = loga y
Laws of logarithms: log(A × B) = log A + log B A log = log A − log B B
b
(iii) Trapezium
1 Area = (a + b)h 2
log An = n × log A
a
Exponential series: ex = 1 + x +
x2 x3 + + ··· 2! 3!
h
(valid for all values of x) b
Theorem of Pythagoras: (iv) Triangle
b2 = a 2 + c2
Area =
1 ×b×h 2
A
c
B
b h
a
C b
List of formulae Area = πr 2 Circumference = 2πr
(v) Circle
(iii)
Pyramid If area of base = A and perpendicular height = h then:
r
s
u
Volume =
1 × A×h 3
r
Radian measure:
2π radians = 360 degrees
h
For a sector of circle: θ◦ (2πr ) = r θ 360
(θ in rad)
1 θ◦ (πr 2 ) = r 2 θ 360 2
(θ in rad)
s=
arc length,
shaded area =
Equation of a circle, centre at origin, radius r :
Total surface area = sum of areas of triangles forming sides + area of base (iv)
Cone 1 Volume = πr2 h 3 Curved surface area = πrl
x 2 + y2 = r 2 Equation of a circle, centre at (a, b), radius r : (x − a)2 + (y − b)2 = r 2
Total surface area = πrl + πr2
Volumes and surface areas of regular solids: (i)
l
Rectangular prism (or cuboid)
h
Volume = l × b × h Surface area = 2(bh + hl + lb)
r
(v)
l
h
Sphere 4 Volume = πr3 3 Surface area = 4πr2
b
(ii)
Cylinder Volume = πr2 h Total surface area = 2πrh + 2πr2 r
r
h
419
420 Basic Engineering Mathematics Areas of irregular figures by approximate methods: Trapezoidal rule width of 1 first + last Area ≈ interval 2 ordinate + sum of remaining ordinates Mid-ordinate rule Area ≈ (width of interval)(sum of mid-ordinates) Simpson’s rule 1 width of first + last Area ≈ ordinate 3 interval sum of even sum of remaining +4 +2 ordinates odd ordinates
For a general sinusoidal function y = A sin (ωt ± α), then A = amplitude ω = angular velocity = 2π f rad/s ω = frequency, f hertz 2π 2π = periodic time T seconds ω α = angle of lead or lag (compared with y = A sin ωt)
Cartesian and polar co-ordinates: If co-ordinate (x, y) = (r, θ ) then y r = x 2 + y 2 and θ = tan−1 x If co-ordinate (r, θ ) = (x, y) then x = r cos θ and y = r sin θ
Mean or average value of a waveform: area under curve length of base sum of mid-ordinates = number of mid-ordinates
mean value, y =
Triangle formulae: Sine rule: Cosine rule:
a b c = = sin A sin B sin C a 2 = b2 + c2 − 2bc cos A
B
b
a
The nth term is: a + (n − 1)d n Sum of n terms, Sn = [2a + (n − 1)d] 2
Geometric progression: If a = first term and r = common ratio, then the geometric progression is: a, ar, ar2 , . . . The nth term is: arn−1 a (1 − r n ) a (r n − 1) or (1 − r) (r − 1)
If − 1 < r < 1, S∞ =
a (1 − r)
C
Area of any triangle 1 × base × perpendicular height 2 1 1 1 ac sin B or bc sin A = ab sin C or 2 2 2 a+b+c = [s (s − a) (s − b) (s − c)] where s = 2 =
If a = first term and d = common difference, then the arithmetic progression is: a, a + d, a + 2d, . . .
Sum of n terms, Sn =
A c
Arithmetic progression:
Statistics: Discrete data:
mean, x¯ =
x
n
2
− x) ¯ (x standard deviation, σ = n
List of formulae Grouped data:
fx mean, x¯ = f
f (x − x) ¯ 2 standard deviation, σ = f
Standard integrals y ax n cos ax
Standard derivatives
sin ax eax
y or f(x)
dy = or f (x) dx
ax n
anx n−1
sin ax
a cosax
cos ax
−a sin ax
eax
aeax
ln ax
1 x
1 x
a
y dx x n+1 + c (except when n = −1) n+1
1 sin ax + c a 1 − cos ax + c a 1 ax e +c a ln x + c
For a copy of List of formulae, go to website: www.routledge.com/cw/bird
421
Areas of plane figures:
a m × a n = a m+n a m/n =
√ n m a
am an
= a m−n (a m )n = a mn
a −n =
a0 = 1 b
Quadratic formula: If ax 2 + bx + c = 0
1 an
Area = l × b
(i) Rectangle
√ −b ± b2 − 4ac x= 2a
then
Equation of a straight line:
l
(ii) Parallelogram
Area = b × h
y = mx + c
Definition of a logarithm: If y = a x
then
h
x = loga y
Laws of logarithms: log(A × B) = log A + log B A log = log A − log B B
b
(iii) Trapezium
1 Area = (a + b)h 2
log An = n × log A
a
Exponential series: ex = 1 + x +
x2 x3 + + ··· 2! 3!
h
(valid for all values of x) b
Theorem of Pythagoras: (iv) Triangle
b2 = a 2 + c2
Area =
1 ×b×h 2
A
c
B
b h
a
C b
List of formulae Area = πr 2 Circumference = 2πr
(v) Circle
(iii)
Pyramid If area of base = A and perpendicular height = h then:
r
s
u
Volume =
1 × A×h 3
r
Radian measure:
2π radians = 360 degrees
h
For a sector of circle: θ◦ (2πr ) = r θ 360
(θ in rad)
1 θ◦ (πr 2 ) = r 2 θ 360 2
(θ in rad)
s=
arc length,
shaded area =
Equation of a circle, centre at origin, radius r :
Total surface area = sum of areas of triangles forming sides + area of base (iv)
Cone 1 Volume = πr2 h 3 Curved surface area = πrl
x 2 + y2 = r 2 Equation of a circle, centre at (a, b), radius r : (x − a)2 + (y − b)2 = r 2
Total surface area = πrl + πr2
Volumes and surface areas of regular solids: (i)
l
Rectangular prism (or cuboid)
h
Volume = l × b × h Surface area = 2(bh + hl + lb)
r
(v)
l
h
Sphere 4 Volume = πr3 3 Surface area = 4πr2
b
(ii)
Cylinder Volume = πr2 h Total surface area = 2πrh + 2πr2 r
r
h
419
420 Basic Engineering Mathematics Areas of irregular figures by approximate methods: Trapezoidal rule width of 1 first + last Area ≈ interval 2 ordinate + sum of remaining ordinates Mid-ordinate rule Area ≈ (width of interval)(sum of mid-ordinates) Simpson’s rule 1 width of first + last Area ≈ ordinate 3 interval sum of even sum of remaining +4 +2 ordinates odd ordinates
For a general sinusoidal function y = A sin (ωt ± α), then A = amplitude ω = angular velocity = 2π f rad/s ω = frequency, f hertz 2π 2π = periodic time T seconds ω α = angle of lead or lag (compared with y = A sin ωt)
Cartesian and polar co-ordinates: If co-ordinate (x, y) = (r, θ ) then y r = x 2 + y 2 and θ = tan−1 x If co-ordinate (r, θ ) = (x, y) then x = r cos θ and y = r sin θ
Mean or average value of a waveform: area under curve length of base sum of mid-ordinates = number of mid-ordinates
mean value, y =
Triangle formulae: Sine rule: Cosine rule:
a b c = = sin A sin B sin C a 2 = b2 + c2 − 2bc cos A
B
b
a
The nth term is: a + (n − 1)d n Sum of n terms, Sn = [2a + (n − 1)d] 2
Geometric progression: If a = first term and r = common ratio, then the geometric progression is: a, ar, ar2 , . . . The nth term is: arn−1 a (1 − r n ) a (r n − 1) or (1 − r) (r − 1)
If − 1 < r < 1, S∞ =
a (1 − r)
C
Area of any triangle 1 × base × perpendicular height 2 1 1 1 ac sin B or bc sin A = ab sin C or 2 2 2 a+b+c = [s (s − a) (s − b) (s − c)] where s = 2 =
If a = first term and d = common difference, then the arithmetic progression is: a, a + d, a + 2d, . . .
Sum of n terms, Sn =
A c
Arithmetic progression:
Statistics: Discrete data:
mean, x¯ =
x
n
2
− x) ¯ (x standard deviation, σ = n
List of formulae Grouped data:
fx mean, x¯ = f
f (x − x) ¯ 2 standard deviation, σ = f
Standard integrals y ax n cos ax
Standard derivatives
sin ax eax
y or f(x)
dy = or f (x) dx
ax n
anx n−1
sin ax
a cosax
cos ax
−a sin ax
eax
aeax
ln ax
1 x
1 x
a
y dx x n+1 + c (except when n = −1) n+1
1 sin ax + c a 1 − cos ax + c a 1 ax e +c a ln x + c
For a copy of List of formulae, go to website: www.routledge.com/cw/bird
421
