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Alright, it's 1:42 am here on the East Coast... I've been working on a set of problems for my Math Physics class for the whole weekend. Four have got me stuck and well, I really need to get them done in order to pass this class (I'm not doing so hot) Thing is, all this is Calc C, and I haven't taken Calc C. I'm just praying you guys can help.

1A. Find the volume of the region bounded by the parabolic cylinder z=4-x^2 and the planes x=0, y=6, y=0, and z=0. Use rectangular coordinates. Make an accurate sketch of the geometry involved.

B. Find the coordinates of the centroid of the above volume.

Where I got stuck here: Well, I know I have to intergrate something, but what? And how many times? I'm not used to 3d geometry.

2. Calculate the moment of inertia, I(z), of that portion of the ellipsoid with surface x^2 + y^2 + z = 4, lying above the plane z=0, with mass density ρ = (1-r^(-2)). Use cylindrical coordinates.

Where I got stuck here: I know was moment of inertia is, that isn't that bad. Just what's a ellipsoid and how the heck do I use cylindrical coordinates?

3. Give that x = r cos φ sin θ , y = r sin θ sin φ , and z = r cos θ , derive the expression for the Jacobian of the transformation from the varibles x, y, & z to r, θ, and φ by Jacobi's method.

What went wrong here: I don't know Jacobi's method and looking it up on the internet didn't help any... and yes I tried MathWorld. :)

4. Calculate the electric potential at the point x = 4 meters, y = 8 meters due to a linear charge density along a rod located between z' = 0 and z' = b = 10 meters. The charge density is 2+(z')^2 C/m. Work out the solution in terms of the variables. Substitute the values of x, y, and b only at the end.

What went wrong here: I know electric potential, that isn't hard. but the 3d geometry is screwing me over again :(

Honestly, thank you thank you thank you for ANY and all help :)

1A. Find the volume of the region bounded by the parabolic cylinder z=4-x^2 and the planes x=0, y=6, y=0, and z=0. Use rectangular coordinates. Make an accurate sketch of the geometry involved.

B. Find the coordinates of the centroid of the above volume.

Where I got stuck here: Well, I know I have to intergrate something, but what? And how many times? I'm not used to 3d geometry.

2. Calculate the moment of inertia, I(z), of that portion of the ellipsoid with surface x^2 + y^2 + z = 4, lying above the plane z=0, with mass density ρ = (1-r^(-2)). Use cylindrical coordinates.

Where I got stuck here: I know was moment of inertia is, that isn't that bad. Just what's a ellipsoid and how the heck do I use cylindrical coordinates?

3. Give that x = r cos φ sin θ , y = r sin θ sin φ , and z = r cos θ , derive the expression for the Jacobian of the transformation from the varibles x, y, & z to r, θ, and φ by Jacobi's method.

What went wrong here: I don't know Jacobi's method and looking it up on the internet didn't help any... and yes I tried MathWorld. :)

4. Calculate the electric potential at the point x = 4 meters, y = 8 meters due to a linear charge density along a rod located between z' = 0 and z' = b = 10 meters. The charge density is 2+(z')^2 C/m. Work out the solution in terms of the variables. Substitute the values of x, y, and b only at the end.

What went wrong here: I know electric potential, that isn't hard. but the 3d geometry is screwing me over again :(

Honestly, thank you thank you thank you for ANY and all help :)

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