Chapter 7 - Applications of Integration - 7.6 Exercises - Page 494: 29

$$\left( {\overline x ,\overline y } \right) \approx \left( {0,{\text{16}}{\text{.18}}} \right)$$

$$\eqalign{ & y = 5\root 3 \of {400 - {x^2}} ,{\text{ }}y = 0 \cr & {\text{Using the graph shown below}} \cr & {\text{Let }}f\left( x \right) = 5\root 3 \of {400 - {x^2}} {\text{ and }}g\left( x \right) = 0,{\text{ on }}\left[ { - 20,20} \right] \cr & \cr & {\text{*The mass of the lamina is }} \cr & m = \rho \int_a^b {\left[ {f\left( x \right) - g\left( x \right)} \right]} dx \cr & m = \rho \int_{ - 20}^{20} {5\root 3 \of {400 - {x^2}} } dx \cr & {\text{Using a graphing utility to approximate the integral}} \cr & m \approx \rho \left( {1239.76} \right) \cr & m \approx 1239.76\rho \cr & \cr & *{\text{The moment about the }}x{\text{ - axis is}} \cr & {M_x} = \rho \int_a^b {\left[ {\frac{{f\left( x \right) + g\left( x \right)}}{2}} \right]} \left[ {f\left( x \right) - g\left( x \right)} \right]dx \cr & {M_x} = \rho \int_0^4 {\left[ {\frac{{5\root 3 \of {400 - {x^2}} }}{2}} \right]} \left[ {5\root 3 \of {400 - {x^2}} } \right]dx \cr & {M_x} = \frac{{25}}{2}\rho {\int_{ - 20}^{20} {\left( {\root 3 \of {400 - {x^2}} } \right)} ^2}dx \cr & {\text{Using a graphing utility to approximate the integral}} \cr & {M_x} \approx \frac{{25}}{2}\rho \left( {1605.141} \right) \cr & {M_x} \approx 20064.27\rho \cr & \cr & *{\text{The moment about the }}y{\text{ - axis is}} \cr & {M_y} = \rho \int_a^b x \left[ {f\left( x \right) - g\left( x \right)} \right]dx \cr & {M_y} = \rho \int_{ - 20}^{20} x \left[ {5\root 3 \of {400 - {x^2}} } \right]dx \cr & {M_y} = 5\rho \int_{ - 20}^{20} {x\root 3 \of {400 - {x^2}} } dx \cr & {M_y} = 5\rho \left( 0 \right) \cr & {M_y} = 0 \cr & \cr & *{\text{The center of mass }}\left( {\overline x ,\overline y } \right){\text{ is given by:}} \cr & \overline x = \frac{{{M_y}}}{m} = \frac{0}{{2.376\rho }} = 0 \cr & \overline y = \frac{{{M_x}}}{m} = \frac{{20064.27\rho }}{{1239.76\rho }} \approx 16.18 \cr & \left( {\overline x ,\overline y } \right) \approx \left( {0,{\text{16}}{\text{.18}}} \right) \cr} $$