5.
(a) Evaluate the surface integral
\iint_\sigma xz \,dS
where \(\sigma\) is the cylindrical surface \(x^2 + y^2 = 1\) bounded by the
planes \(x = 0\), \(z = 0\) and \(z = 1\), and the normal vector pointing
outward from the surface.
[9 marks]
(b) Given the surface integral
\iint_\sigma \mathbf{F} \cdot \mathbf{n} \,dS
where \(\mathbf{F} = xy^2 \mathbf{i} + x^2y \mathbf{j} + 4z \mathbf{k}\), \(\sigma\) is the surface of the cylinder
defined by \(x^2 + y^2 \le 4\) such that \(0 \le z \le 4\), and \(\mathbf{n}\) is the out-
ward unit normal vector to the surface \(\sigma\). Use Gauss' Theorem to
evaluate the integral.
[6 marks]
(c) A tetrahedron is bounded by the planes \(x + 2y + 4z = 4\), \(x = 0\),
\(y = 0\), and \(z = 0\). The vector field
\(\mathbf{F} = x^2 \mathbf{i} + xyz \mathbf{j} + 4z \mathbf{k}\),
is defined on the open surface \(\sigma\) which is part of the tetrahedron
but not including the plane \(x + 2y + 4z = 4\). The orientation of the
surface \(\sigma\) is outward with \(C\) (the closed curve on the boundary)
oriented clockwise as viewed from above. Using Stokes' Theorem,
evaluate the integral
\iint_\sigma (\nabla \times \mathbf{F}) \cdot \mathbf{n} \,dS.
[5 marks]
