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Example 4 _ Plane Areas in Rectangular Coordinates _ Integral Calculus Review
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Discussion part for experiment properties of alkenesFull description
D Properties of Plane Areas
Notation:
y
1
Rectangle
h
C
y
x
(Origin of axes at centroid) b
x 2
A bh
x
A area x , y distances to centroid C I x , I y moments of inertia with respect to the x and y axes, respectively I xy product of inertia with respect to the x and y axes I P I x I y polar moment of inertia with respect to the origin of the x and y axes I BB moment of inertia with respect to axis B-B
bh3 I x 12
h
y 2
hb3 I y 12
I xy 0
bh I P (h2 b2) 12
b y
2
Rectangle
B
(Origin of axes at corner)
bh3 I x 3 h O B
3
I BB
x
b2h2 I xy 4
bh I P (h2 + b2) 3
3
b h 6(b h ) 2
2
b y
3
Triangle c
bh A 2
x h C y b
966
hb3 I y 3
x
bh3 I x 36
(Origin of axes at centroid) bc
x 3
h
y 3
bh I y (b2 bc c2) 36
bh2 I xy (b 2c) 72
bh I P (h2 b2 bc c2) 36
APPENDIX D
y
4
Triangle
c B
B h
O
x
b y
5
bh3 I x 12
(Origin of axes at vertex) bh I y (3b2 3bc c2) 12
bh2 I xy (3b 2c) 24
Isosceles triangle bh A 2
x h C
y
B
x B
b
bh3 I x 36
bh3 I BB 4
(Origin of axes at centroid)
b
x 2
h
y 3
hb3 I y 48
I xy 0 bh3 I BB 12
bh I P (4h2 3b2) 144
( Note: For an equilateral triangle, h 3 b /2.) y
6
x
h
C
y
B
x B
b
B
b
bh3 I x 36
bh3 I x 12
h
x 3
x
h
y 3
hb3 I y 36
b2h2 I xy 72 bh3 I BB 12
(Origin of axes at vertex)
hb3 I y 12
bh I P (h2 b2) 12
b2h2 I xy 24 bh3 I BB 4
b y
8
Trapezoid
a
h
bh A 2
Right triangle
B
O
(Origin of axes at centroid)
bh I P (h2 b2) 36
y
7
Right triangle
C
h(a b) A 2 y
x B
B b
(Origin of axes at centroid)
I x
h(2a b) y 3(a b)
h3(a2 4ab b2) 36(a b)
I BB
h3(3a b)
1 2
Properties of Plane Areas
967
968
APPENDIX D
9
y
Properties of Plane Areas
Circle
d = 2r
p d 2 A p r 2 4
r x
C B
I xy
B y
10
r
0
I P
Semicircle
C y
x
B
2
I x
y
B
y
B x
O
I BB
4
5p r 4 5p d 4 4 64
p r 4 I y 8
(9p 64)r 0.1098r 72p
p r 2 A 4 C
p r 4 p d 4 2 32
4r y 3p 4
Quarter circle x
p r 4 p d 4 I x I y 4 64
(Origin of axes at centroid)
p r 2 A 2
B 11
(Origin of axes at center)
I xy 0
p r 4 I BB 8
(Origin of axes at center of circle)
4r x y 3p
p r 4 I x I y 16
r 4 I xy 8
2
I BB
4
(9p 64)r 0.05488r 144p
4
r y
12
Quarter-circular spandrel
B
B
r x C
y x
O y
13
x
a
x
A a r 2
C a x
(10 3p )r y 3(4 p ) 0.2234r
2r 0.7766r 3(4 p )
5p I x 1 r 4 0.01825r 4 16
I y I BB
1 3
p 4 r 0.1370r 4 16
(Origin of axes at center of circle)
a angle in radians
r O
Circular sector x
y
p A 1 r 2 4
(Origin of axes at point of tangency)
(a p /2)
x r sin a
r 4 I x (a sin a cos a ) 4
2r sin a y 3a r 4 I y (a sin a cos a ) 4
I xy 0
a r 4 I P 2
APPENDIX D
Circular segment
y
14
y
a
(Origin of axes at center of circle)
a angle in radians
C
(a p /2)
2r sin3 a y 3 a sin a cos a
A r 2(a sin a cos a )
a r
r 4 I x (a sin a cos a 2 sin3 a cos a ) 4
x
O
Properties of Plane Areas
I xy 0
r 4 I y (3a 3 sin a cos a 2 sin3 a cos a ) 12 15
a
y
Circle with core removed a angle in radians
r
a
C
a a arccos r
b x
a b
(Origin of axes at center of circle)
(a p /2)
3ab r 4 I x 3a 6 r 2
2ab3 r
4
ab A 2r 2 a r 2
b r 2 a 2
2ab3 r 4 ab I y a 2 r 2 r 4
2a y
16
Ellipse A p ab
b C
x b
a
(Origin of axes at centroid)
I xy 0
a
p a b 3 I x 4
p b a 3 I y 4
p ab I P (b2 a2) 4
Circumference p [1.5( a
b) a b]
4.17b / a 4a 2
17 y
Parabolic semisegment
ertex V y = f ( x )
x
h
C O
y b
x
(0 b a /3)
(Origin of axes at corner)
x 2 y f ( x ) h 1 b2
2 bh A 3 16bh3 I x 105
3b x 8
( a /3 b a)
2h y 5
2hb3 I y 15
b2h2 I xy 12
I xy 0
969
970
APPENDIX D
18
Properties of Plane Areas
Parabolic spandrel
y = f ( x ) x
h x 2 y f ( x ) b2
h
ertex V
y
C
O
3b x 4
bh 3
x A b
bh3 I x 21 y
19
hb3 I y 5
x n y f ( x ) h 1 bn
x h
C
y
b2h2 I xy 12
b
I x
y
n
x
n1
3
h x n y f ( x ) bn
x
h C
y
O
21
I x
3
h
C
x B I
y
B b
b
x
y
d = 2r
t
I xy 0
b2h2 I xy 4(n 1)
p h
y 8
8bh3 I BB 9p
p d 3t I x I y p r 3t 8
p d 3t I P 2p r 3t 4
4 32 I y hb3 0.2412hb3 p p 3
Thin circular ring (Origin of axes at center) Approximate formulas for case when t is small
x
C
2
2
b h n 4(n 1)(n 2)
(Origin of axes at centroid)
A 2p r t p dt
r
2
I xy
h(n 1) y 2(2n 1)
8 p 3 3 bh 0.08659bh 9p 16
I xy 0
hb3n I y 3(n 3)
(Origin of axes at point of tangency)
hb3 I y n3
4bh A p
hn
y 2n 1
(n 0)
bh 3(3n 1)
Sine wave
y
b(n 1) 2(n 2)
b(n 1) x n2
bh A n1 x
b
(n 0)
3
Spandrel of nth degree y = f ( x )
(Origin of axes at corner)
2bh n (n 1)(2n 1)(3n 1)
A bh x
O
22
3h y 10
Semisegment of nth degree y = f ( x )
20
(Origin of axes at vertex)
APPENDIX D
23
t B
C b y
b
b angle in radians
x
O
I x r 3t (b sin b cos b )
2b sin2b 2
1 cos2b b
Thin rectangle (Origin of axes at centroid) Approximate formulas for case when t is small A bt
b
b C
x t
B
I y r 3t ( b si n b co s b )
I BB r 3t
I xy 0 y
( Note: For a semicircular arc, b p /2.)
r sin b y b
A 2b rt
r
24
971
Thin circular arc (Origin of axes at center of circle) Approximate formulas for case when t is small
y
B
Properties of Plane Areas
t b3 I x sin2 b 12
t b3 I y cos2 b 12
tb3 I BB sin2 b 3
B
Regular polygon with n sides
25
A b R1
(Origin of axes at centroid)
B
C centroid (at center of polygon)
b
n number of sides (n 3)
R2
b length of a side
b central angle for a side C a
360° b n
a
a interior angle (or vertex angle)
n2 180° n
a b 180°
R1 radius of circumscribed circle (line CA) b b R1 csc 2 2
b b R2 cot 2 2
R2 radius of inscribed circle (line CB)
b nb2 A cot 4 2
I c moment of inertia about any axis through C (the centroid C is a principal point and every axis through C is a principal axis)
