Answer to Question #161475 in Calculus for Mano

Question #161475

Find the mass of a triangular lamina whose vertices are (0,0), (1, 1), (0.1). if the density of the lamina is p(x, y) = sin (y^2).


1
Expert's answer
2021-02-24T07:59:19-0500

Consider the triangular lamina as shown in the figure below:






The region RR in cartesian coordinates is given by,


R={(x,y)0y1,0xy}R=\{(x,y)|0\eqslantless y \eqslantless1, 0 \eqslantless x \eqslantless y \}


The mass of the lamina with density ρ(x,y)=sin(y2)\rho(x,y)=sin(y^2) is evaluated as,


Mass(m)=Rρ(x,y)dA(m)=\iint_{R}\rho(x,y)dA


=y=01x=0ysin(y2)dxdy=\int_{y=0}^{1}\int_{x=0}^{y}sin(y^2)dxdy


=y=01sin(y2)[x]x=0ydy=\int_{y=0}^{1}sin(y^2)[x]_{x=0}^{y}dy


=y=01ysin(y2)dy=\int_{y=0}^{1}ysin(y^2)dy


Let u=y2u=y^2 , then 12du=ydy\frac{1}{2}du=ydy , 0u10\eqslantless u \eqslantless 1 , so the integral is,


=u=0112sin(u)du=\int_{u=0}^{1}\frac{1}{2}sin(u)du


=12[cos(u)]=01=\frac{1}{2}[-cos(u)]_{=0}^{1}


=12(1cos(1))0.22984=\frac{1}{2}(1-cos(1)) \approx 0.22984


Therefore, the mass of the lamina is m=12(1cos(1))0.22984m=\frac{1}{2}(1-cos(1)) \approx 0.22984 .

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