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زمان تقریبی مطالعه: 3 دقیقه
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فهرست مرکزهای هندسی

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نامنگاره x ¯ {\displaystyle {\bar {x}}}
y ¯ {\displaystyle {\bar {y}}}
مساحت
مثلث راست‌گوشه
− b 3 {\displaystyle {\frac {-b}{3}}}
h 3 {\displaystyle {\frac {h}{3}}}
b h 2 {\displaystyle {\frac {bh}{2}}}
ربع دایره
4 r 3 π {\displaystyle {\frac {4r}{3\pi }}}
4 r 3 π {\displaystyle {\frac {4r}{3\pi }}}
π r 2 4 {\displaystyle {\frac {\pi r^{2}}{4}}}
نیم‌دایره | align="center"| 0 {\displaystyle \,\!0}
0 4 r 3 π {\displaystyle {\frac {4r}{3\pi }}}
ربع بیضی | align="center"| 4 a 3 π {\displaystyle {\frac {4a}{3\pi }}}
4 b 3 π {\displaystyle {\frac {4b}{3\pi }}}
π a b 4 {\displaystyle {\frac {\pi ab}{4}}}
نیم‌بیضی
0 {\displaystyle \,\!0}
4 b 3 π {\displaystyle {\frac {4b}{3\pi }}}
π a b 2 {\displaystyle {\frac {\pi ab}{2}}}
نیمه سهمی گون The area between the curve y = h b 2 x 2 {\displaystyle y={\frac {h}{b^{2}}}x^{2}}
and the y {\displaystyle \,\!y}
axis, from x = 0 {\displaystyle \,\!x=0}
to x = b {\displaystyle \,\!x=b}
3 b 8 {\displaystyle {\frac {3b}{8}}}
3 h 5 {\displaystyle {\frac {3h}{5}}}
2 b h 3 {\displaystyle {\frac {2bh}{3}}}
سهمی The area between the curve y = h b 2 x 2 {\displaystyle \,\!y={\frac {h}{b^{2}}}x^{2}}
and the line y = h {\displaystyle \,\!y=h}
0 {\displaystyle \,\!0}
3 h 5 {\displaystyle {\frac {3h}{5}}}
4 b h 3 {\displaystyle {\frac {4bh}{3}}}
Parabolic spandrel The area between the curve y = h b 2 x 2 {\displaystyle \,\!y={\frac {h}{b^{2}}}x^{2}}
and the x {\displaystyle \,\!x}
axis, from x = 0 {\displaystyle \,\!x=0}
to x = b {\displaystyle \,\!x=b}
3 b 4 {\displaystyle {\frac {3b}{4}}}
3 h 10 {\displaystyle {\frac {3h}{10}}}
b h 3 {\displaystyle {\frac {bh}{3}}}
General spandrel The area between the curve y = h b n x n {\displaystyle y={\frac {h}{b^{n}}}x^{n}}
and the x {\displaystyle \,\!x}
axis, from x = 0 {\displaystyle \,\!x=0}
to x = b {\displaystyle \,\!x=b}
n + 1 n + 2 b {\displaystyle {\frac {n+1}{n+2}}b}
n + 1 4 n + 2 h {\displaystyle {\frac {n+1}{4n+2}}h}
b h n + 1 {\displaystyle {\frac {bh}{n+1}}}
قاچ دایره The area between the curve (in polar coordinates) r = ρ {\displaystyle \,\!r=\rho }
and the pole, from θ = − α {\displaystyle \,\!\theta =-\alpha }
to θ = α {\displaystyle \,\!\theta =\alpha }
2 ρ sin ⁡ ( α ) 3 α {\displaystyle {\frac {2\rho \sin(\alpha )}{3\alpha }}}
0 {\displaystyle \,\!0}
α ρ 2 {\displaystyle \,\!\alpha \rho ^{2}}
قطعه دایره
0 {\displaystyle \,\!0}
4 R sin 3 ⁡ θ 2 3 ( θ − sin ⁡ θ ) {\displaystyle {\frac {4R\sin ^{3}{\frac {\theta }{2}}}{3(\theta -\sin {\theta })}}}
R 2 2 ( θ − s i n θ ) {\displaystyle {\frac {R^{2}}{2}}(\theta -sin{\theta })}
Quarter-circular arc The points on the circle x 2 + y 2 = r 2 {\displaystyle \,\!x^{2}+y^{2}=r^{2}}
and in the first quadrant 		2 r π {\displaystyle {\frac {2r}{\pi }}}
2 r π {\displaystyle {\frac {2r}{\pi }}}
π r 2 {\displaystyle {\frac {\pi r}{2}}}
کمان نیم دایره The points on the circle x 2 + y 2 = r 2 {\displaystyle \,\!x^{2}+y^{2}=r^{2}}
and above the x {\displaystyle \,\!x}
axis 		0 {\displaystyle \,\!0}
4 r π 3 {\displaystyle {\frac {4r}{\pi 3}}}
π r {\displaystyle \,\!\pi r}
کمان دایره The points on the curve (in polar coordinates) r = ρ {\displaystyle \,\!r=\rho }
ØŒ from θ = − α {\displaystyle \,\!\theta =-\alpha }
to θ = α {\displaystyle \,\!\theta =\alpha }
ρ sin ⁡ ( α ) α {\displaystyle {\frac {\rho \sin(\alpha )}{\alpha }}}
0 {\displaystyle \,\!0}
2 α ρ {\displaystyle \,\!2\alpha \rho }

منابع

ویکی‌پدیای انگلیسی

پیوند به بیرون

  • http://www.engineering.com/Library/ArticlesPage/tabid/85/articleType/ArticleView/articleId/109/Centroids-of-Common-Shapes.aspx
  • http://www.efunda.com/math/areas/IndexArea.cfm
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