Surfaces and surface integrals of scalar functions

1 Surfaces and surface integrals of scalar functions

Find a parametrization of the portion of the sphere \(x^2 + y^2 + z^2 = 9 \) above the plane \(z = 3/2\). Specify the domain and the expression of the parametrization.
Find a parametrization of the portion of the cylinder \(x^2 + y^2 = 16 \) between the planes \( x+ y + z = 0\) and \(2x + y - z = - 10\). Specify the domain and the expression of the parametrization.
Find a parametrization of the portion of the cone \( z = \sqrt {3 x^2 + 3 y^2 } \) outside the sphere \(x^2 + y^2 + (z- 2)^2 = 4\) and inside the sphere \(x^2 + y^2 + (z- 5)^2 = 25\). Specify the domain and the expression of the parametrization.
Find a parametrization of the portion of the graph \( z = \sin x + \sin y \) between the lines \(x= 0\), \(x = \pi \), \(y = x\), and \(y = \pi \). Specify the domain and the expression of the parametrization.
For each of the previous problems, take your parametrization
\[ \varphi (u,v) = (x(u,v), y(u,v) , z(u,v)) \]
and consider the vector \(\frac {\partial \varphi }{\partial u } \times \frac {\partial \varphi }{\partial v}\). Identify which side of the surface it comes out of. Construct a second parametrization of the same surface where the corresponding vector points in the opposite direction.
Using a parametrization and its Jacobian, find the area of the portion of the plane \(5x + 3y + 4z = 60 \) in the first octant (\(x, y, z \geq 0\)). Assumme a metal lamina has this shape and its density is given by \(\rho (x,y,z) = \sin x + 3\). Find its mass.
Find the area of the portion of the paraboloid \(x = 9 - y^2 - z^2\) with \(x \geq - 16\). Assume a metal lamina has this shape and its density is given by \(\rho (x,y,z) = x - y + 21\). Find its mass.
Find
\[ \iint _{\Sigma } (z^2 +x + 3 ) \, dS \]
where \(\Sigma \) is the portion of the sphere \( x^2 + y^2 + z^2 = 16\) above the plane \(z = -2\).
Find
\[ \iint _{\Sigma } ( x^2 + 3 x + 2 y^2 - 2y + z^2 + 3 ) \, dS \]
where \(\Sigma \) is the portion of the cone \( z = \sqrt {3 x^2 + 3 y^2 } \) outside the sphere \(x^2 + y^2 + (z- 2)^2 = 4\) and inside the sphere \(x^2 + y^2 + (z- 5)^2 = 25\).
Consider the surface \(\Sigma \subset \mathbb {R}^3\) with parametrization
\[ \varphi : [0, 2 \pi ] \times [0, 2 \pi ] \to \mathbb {R}^3 \]
given by
\[ \varphi (u,v) = ( (3 + \cos v) \cos u , (3 + \cos v) \sin u , \sin v ) \]
Describe the surface \(\Sigma \) and find its area. Compute
\[ \iint _{\Sigma } ( x + 2y + z^2 +3 ) \, dS \]
Compute
\[ \iint _{\Sigma } ( xz + y ) \, dS \]
where \(\Sigma \subset \mathbb {R}^3\) is the helicoid with parametrization
\[ \varphi (u,v) = ( u \cos v, u \sin v , v ) \]
with \(1 \leq u \leq 3\), \(0 \leq v \leq 4 \pi \).