Question

Surface Integrals Prolbem? If the mass per unit area of a surface is given by ρ=xy, find the mass if S is the part of the cylinder x^2+z^2=25 which is in the first octant and contained within the cylinder x^2+y^2=16.

So here was my attempt.
I parametrized the curve.
x^2+z^2=25
r(u, v) = <5cos(u), v, 5sin(u)>
I then plugged into the...

Answer 1

To find the mass, we need to evaluate the surface integral of the mass density function ρ=xy over the given surface S. The surface is defined by two equations: x^2+z^2=25 and x^2+y^2=16. We can parametrize the surface using cylindrical coordinates and evaluate the integral to find the mass.

Step-by-step explanation:

To find the mass, we need to evaluate the surface integral of the mass density function ρ=xy over the given surface S. The surface is defined by two equations: x^2+z^2=25 and x^2+y^2=16. We can parametrize the surface using cylindrical coordinates as r(u, v) = <5cos(u), v, 5sin(u)> where u is in the range [0,π/2] and v is in the range [0,√16-y^2].

Using the parametrization, we can calculate the cross product of the partial derivatives of r with respect to u and v, which gives us the surface area element dS = √(1+(∂z/∂u)^2+(∂z/∂v)^2)dudv = √(1+25cos^2(u))dudv.

We can now set up the integral ∫∫S ρ dS = ∫∫S xy√(1+25cos^2(u))dudv, where S is the region defined by the given equations, and evaluate it using the parametrization limits and the surface area element.

Answer 2

You're trying to find the mass of the region, so a triple integral is probably more appropriate here.


`\displaystyle\text{mass}_S=\iiint_S\rho(x,y,z)\,\mathrm dV=\int_0^4\int_0^(√(16-x^2))\int_0^(\sqrt25-x^2) xy\,\mathrm dz\,\mathrm dy\,\mathrm dx=\frac{706}5`