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Physics LibreTexts

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
\( y = y(x) \)  \( r = r(θ) \)

\( \overline{x} =  \frac{1}{L}  \int_a^b x[1+( \frac{dy}{dx})^{2}]^{\frac{1}{2}}\)

 

\( \overline{y} =  \frac{1}{L}  \int_a^b y[1+( \frac{dy}{dx})^{2}]^{\frac{1}{2}}\)

\( \overline{x} =  \frac{1}{L}  \int_ \alpha ^ \beta  rcos \theta [( \frac{dr}{d \theta })^{2} + r^{2} ]^ \frac{1}{2} \)

 

\( \overline{y} =  \frac{1}{L}  \int_ \alpha ^ \beta  rsin \theta [( \frac{dr}{d \theta })^{2} + r^{2} ]^ \frac{1}{2} \)

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