Surface integrals of vector fields

1 Surface integrals of vector fields

Let \(\Sigma \) be the portion of the plane \(2x + y + z = 6\) inside the cylinder \(x^2 + y^2 = 4\), oriented upwards. Compute
\[ \iint _{\Sigma } \langle z, y, 3 - x \rangle \cdot dS \]
Let \(\Sigma \) be the portion of the plane \(z = 3x + 2y + 1\) between the planes \(x= 0 \) and \(x=1\) and between the planes \(y = 0 \) and \(y= 2\), oriented upwards. Compute
\[ \iint _{\Sigma } \langle y + 2, z - 3 , x + 1 \rangle \cdot dS \]
Let \(\Sigma \) be the triangle with vertices \((1,0,0)\), \((0,2,0)\), \((0,0,3)\), oriented downwards. Compute
\[ \iint _{\Sigma } \langle x + 2, 2y - 5, 3z + 1 \rangle \cdot dS \]
Let \(\Sigma \) be the portion of the paraboloid \(z = 4 - x^2 - y^2\) above the plane \(z = 0\), oriented upwards. Compute
\[ \iint _{\Sigma } \langle x, y, 0 \rangle \cdot dS \]
Let \(\Sigma \) be the portion of the cone \(z = \sqrt {x^2 + y^2}\) between the planes \(z=1\) and \(z=3\), oriented outward (away from the axis). Compute
\[ \iint _{\Sigma } \langle z, 3 + x, 1 - y \rangle \cdot dS \]
Let \(\Sigma \) be the portion of the sphere \(x^2 + y^2 + z^2 = 9\) with \(z \geq 0\), oriented upward. Compute
\[ \iint _{\Sigma } \langle x, y, z \rangle \cdot dS \]
Let \(\Sigma \) be the portion of the paraboloid \(y = x^2 + z^2\) inside the cylinder \(x^2 + z^2 = 1\), oriented in the direction of the \(y\)-axis. Compute
\[ \iint _{\Sigma } \langle y , x + 1 , z \rangle \cdot dS \]
Let \(\Sigma \) be the portion of the cone \(x = \sqrt {3 y^2 + 3 z^2}\) with \(x \leq \frac {\sqrt {3}}{2}\) , oriented in the direction of the \(x\)-axis. Compute
\[ \iint _{\Sigma } \langle 2, 3x - 1 , y + z \rangle \cdot dS \]