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# 1.5: Summary of the Formulas for Plane Laminas and Curves

## Uniform Plane Lamina

 $$y = y(x)$$ $$r = r(θ)$$ $$\overline{x} = \frac{1}{A} \int_a^b xydx$$ $$\overline{y} = \frac{1}{2A} \int_a^b y^{2}dx$$ $$\overline{x} = \frac{2 \int_ \alpha ^ \beta r^3 cos \theta d \theta }{3 \int_ \alpha ^ \beta r^2 d \theta }$$ $$\overline{y} = \frac{2 \int_ \alpha ^ \beta r^3 sin \theta d \theta }{3 \int_ \alpha ^ \beta r^2 d \theta}$$

## Uniform Plane Curve

 $$y = y(x)$$ $$r = r(θ)$$ $$\overline{x} = \frac{1}{L} \int_a^b x[1+( \frac{dy}{dx})^{2}]^{\frac{1}{2}}$$ $$\overline{y} = \frac{1}{L} \int_a^b y[1+( \frac{dy}{dx})^{2}]^{\frac{1}{2}}$$ $$\overline{x} = \frac{1}{L} \int_ \alpha ^ \beta rcos \theta [( \frac{dr}{d \theta })^{2} + r^{2} ]^ \frac{1}{2}$$ $$\overline{y} = \frac{1}{L} \int_ \alpha ^ \beta rsin \theta [( \frac{dr}{d \theta })^{2} + r^{2} ]^ \frac{1}{2}$$