Consider the triangular lamina as shown in the figure below:
The region R in cartesian coordinates is given by,
R={(x,y)∣0⪕y⪕1,0⪕x⪕y}
The mass of the lamina with density ρ(x,y)=sin(y2) is evaluated as,
Mass(m)=∬Rρ(x,y)dA
=∫y=01∫x=0ysin(y2)dxdy
=∫y=01sin(y2)[x]x=0ydy
=∫y=01ysin(y2)dy
Let u=y2 , then 21du=ydy , 0⪕u⪕1 , so the integral is,
=∫u=0121sin(u)du
=21[−cos(u)]=01
=21(1−cos(1))≈0.22984
Comments
Leave a comment