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Homework Statement
I want to know if I've gone about setting up these integrals in these questions properly before I evaluate them.
(i). Find the mass of the cylinder [itex]S: 0 ≤ z ≤ h, x^2 + y^2 ≤ a^2[/itex] if the density at the point (x,y,z) is [itex]δ = 5z^4 + 6(x^2 - y^2)^2[/itex].
(ii). Evaluate the integral of [itex]f(x,y,z) = (x^2 + y^2 + z^2)^{1/2}[/itex] over the region D which is above the cone [itex]z^2 = 3(x^2 + y^2)[/itex] and between the spheres [itex]x^2 + y^2 + z^2 = 1[/itex] and [itex]x^2 + y^2 + z^2 = 9[/itex].
Homework Equations
Cylindrical polars and spherical polars.
The Attempt at a Solution
(i). Switching to cylindrical polars, x=rcosθ, y=rsinθ and z=z.
So [itex]S → S^{'}: 0 ≤ z ≤ h, -a ≤ r ≤ a, 0 ≤ \theta ≤ 2 \pi[/itex]
and the density becomes [itex]δ = 5z^4 + 6r^4cos^2(2θ)[/itex].
The Jacobian of the cylindrical polars is just r so our integral over S' becomes :
[itex]\int_{0}^{2π} \int_{-a}^{a} \int_{0}^{h} 5rz^4 + 6r^5cos^2(2θ) dzdθdr[/itex]
(ii). Switching to spherical polars, x=ρcosθsinφ, y=ρsinθsinφ and z=ρcosφ and we also know that [itex]x^2+y^2+z^2 = ρ^2[/itex].
So f(x,y,z) now becomes [itex](ρ^2)^{1/2} = ρ[/itex]
The two spheres yield [itex]1 ≤ ρ ≤ 3[/itex]
θ is as is in cylindrical, so [itex]0 ≤ θ ≤ 2π[/itex]
Also [itex]0 ≤ φ ≤ π[/itex]
The purpose of the cone is not yet clear to me.