Question

a. Find the center of mass for lamina defined by the interior of the polar curve r=sin(3$\theta$) 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.

1. Find the center of mass for lamina defined by the interior of the polar curve r sin30 with density that varies according to p(r,0) 2. Find the volume of the cylinder (x-2+y inside the sphere y. 3

Answer 1

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)