Finding Center of Mass in Spherical Coordinates | Calculus Problem Solving

[Shay10825]

Hello everyone! I need some help with the following problems.

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2. Use spherical coordinates to find the center of mass of the sphere of radius 7 and centered at the origin in which the density at any point is proportional to the distance of the point from the z-axis.

My work: http://img54.imageshack.us/img54/820/calc26pj.jpg

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3. Use a change of variables to find the volume of the solid region lying above the plane region R bounded below by the parallelogram with vertices (0,0), (-2,3), (2,5) and (4,2) and above the surface z = [(3x + 2y)^2]*[sqrt( 2y – x)]

My work: http://img54.imageshack.us/img54/338/calc37so.jpg

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4. Find the area of the surface of the graph of f(x,y) = 121 – x^2 – y^2 over the region R = {(x,y) : x^2 + y^2 <= 121}

My work: http://img54.imageshack.us/img54/295/calc45mq.jpg

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Any help would be greatly appreciated.

Thanks

 

[mighty2000]

for posting your questions, let's take a look at each of them:

2. To find the center of mass, we need to use the formula:

x = (1/M)∫∫∫xρ(x,y,z)dV
y = (1/M)∫∫∫yρ(x,y,z)dV
z = (1/M)∫∫∫zρ(x,y,z)dV

Where M is the total mass of the sphere, ρ(x,y,z) is the density at any point, and dV is the volume element in spherical coordinates.

First, we need to find the total mass of the sphere. Since the density is proportional to the distance from the z-axis, we can write ρ(x,y,z) = kr, where k is a constant and r is the distance from the z-axis.

The total mass can be found by integrating ρ(x,y,z) over the entire sphere:
M = ∫∫∫ρ(x,y,z)dV = k∫∫∫r dV

Since the sphere has a radius of 7, the limits of integration for r are from 0 to 7. The volume element in spherical coordinates can be written as dV = r^2sinθ dr dθ dϕ.

Substituting these values into the integral, we get:
M = k∫∫∫r^3sinθ dr dθ dϕ
= k∫0^2π∫0^π∫0^7r^3sinθ dr dθ dϕ
= k(2π)(π)(7^4/4)
= 1029πk

Now, we can use the formula for x, y, and z to find the center of mass:
x = (1/M)∫∫∫xρ(x,y,z)dV
= (1/1029πk)∫0^2π∫0^π∫0^7r^5sinθcosϕ dr dθ dϕ
= 0

Similarly, we can find y and z to be 0 as well. Therefore, the center of mass of the sphere is located at the origin.

3. To find the volume of the solid region, we need to use a change of variables. Let u = 3x