Consider a uniform flat plate with density rho = 1 bounded above by the graph of
ID: 2852765 • Letter: C
Question
Consider a uniform flat plate with density rho = 1 bounded above by the graph of y = 1/(1 + x^2), bounded below by the graph of y = -1/(1 + x^2), bounded on the left by the y axis, and with no bound on the right. What is the mass of this plate? For any b > 0, compute the integral integral_0^b 2x/1 + x^2 dx = (your answer will depend on b) We can conclude that M_y = lim b rightarrow infinity 2x/1 + x^2 dx = infinity, so this infinite plate has finite mass, but does not have a finite center of mass.Explanation / Answer
a) mass of plate =2*[0 to infinity] *1/(1+x2) dx
mass of plate =2*[0 to infinity] 1 *1/(1+x2) dx
mass of plate =2*[0 to infinity] 1/(1+x2) dx
mass of plate =2*|[0 to infinity] tan-1x
mass of plate =2* [tan-1 infinity -tan-10]
mass of plate =2* [pi/2 -0]
mass of plate =pi
b)[0 to b] 2x/(1+x2) dx
=[0 to b] ln(1+x2)
=ln(1+b2) -ln(1+02)
=ln(1+b2) -ln(1)
=ln(1+b2) -0
=ln(1+b2)
My=lim b-> infinityln(1+b2)
My=ln(1+ infinity2)
My= infinity
so this infinate plate has finate mass , but does not have a finate centre of mass