1.5: Summary of the Formulas for Plane Laminas and Curves
Uniform Plane Lamina |
| \( y = y(x) \) | \( r = r(θ) \) |
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\( \overline{x} = \frac{1}{A} \int_a^b xydx\)
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\( \overline{x} = \frac{2 \int_ \alpha ^ \beta r^3 cos \theta d \theta }{3 \int_ \alpha ^ \beta r^2 d \theta } \)
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Uniform Plane Curve |
| \( y = y(x) \) | \( r = r(θ) \) |
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\( \overline{x} = \frac{1}{L} \int_a^b x[1+( \frac{dy}{dx})^{2}]^{\frac{1}{2}}\)
\( \overline{y} = \frac{1}{L} \int_a^b y[1+( \frac{dy}{dx})^{2}]^{\frac{1}{2}}\) |
\( \overline{x} = \frac{1}{L} \int_ \alpha ^ \beta rcos \theta [( \frac{dr}{d \theta })^{2} + r^{2} ]^ \frac{1}{2} \)
\( \overline{y} = \frac{1}{L} \int_ \alpha ^ \beta rsin \theta [( \frac{dr}{d \theta })^{2} + r^{2} ]^ \frac{1}{2} \) |