a. Find the center of mass for lamina defined by the interior of the polar curve r=sin(3) with a density that varies according to p(r,theta)=1/r
b. Find the volume of the cylinder inside the sphere
For part a I got a mass of 2 but not sure about the x bar and y bar calculations.
For part b Im stuck on the z bounds for the integral when doing the problem with the cylindrical coordinate method.
The lamina be described usin∫∫(y(((dA∫∫ρ(x,y)dA
∫∫yρ(x,y)dA∫∫ρ(x,y)dAg polar coordinates with the region:
0≤r≤a0≤r≤a
0≤3θ≤3πthe formula for center of mass (y - coordinate in this case) of a 2d object is:
∫∫yρ(x,y)dA∫∫ρ(x,y)dA∫∫yρ(x,y)dA∫∫ρ(x,y)dA
where ρ(x,y)ρ(x,y)=1/r is density.
we need is a function for density, a description of y in polar
coordinates, and a description of dA in polar. The density is
proportion to the distance from origin.
density=kx2+y2−−−−−−√density=kx2+y2
Where k is the proportionality constant. x2+y2=r2x2+y2=r2 in polar coordinates. Hence
density=krdensity=krKr=1/rK=1/r^2
We also know that in polar coordinates y=rcos(θ)y=rcos(θ) and
dA=rdrdθdA=rdrdθ.
given that r=sin3(theta)
denstiy=1/r=1/sin3(theta)=cosec3(theta)
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