Surface Integrals

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Evaluate the surface integral $\iint_{S} x^{2} y z d S$ if $S$ is the part of the plane $z = 1 + 2 x + 3 y$ above the rectangle $[0, 3] \times [0, 2]$ .

$171 \sqrt{14}$
Evaluate the surface integral $\iint_{S} y z d S$ if $S$ is the part of the plane $x + y + z = 1$ that lies in the first octant.

$\frac{\sqrt{3}}{24}$
Evaluate the surface integral $\iint_{S} x^{2} z^{2} d S$ if $S$ is the part of the cone $z^{2} = x^{2} + y^{2}$ that lies between the planes $z = 1$ and $z = 3$ .

$\frac{364 \sqrt{2} π}{3}$
Evaluate the surface integral $\iint_{S} y d S$ if $S$ is the part of the paraboloid $y = x^{2} + z^{2}$ that lies inside the cylinder $x^{2} + z^{2} = 4$ .

$\frac{π (391 \sqrt{17} + 1)}{60}$
Evaluate the surface integral $\iint_{S} (x^{2} z + y^{2} z) d S$ if $S$ is the part of the hemisphere $x^{2} + y^{2} + z^{2} = 4$ above the $x y -plane$ .

$16 π$
Find the flux of $F$ across $S$ if $F (x, y, z) = ⟨x y, y z, z x⟩$ and $S$ is the part of the paraboloid $z = 4 - x^{2} - y^{2}$ that lies above the square $0 \leq x \leq 1$ , $0 \leq y \leq 1$ , and has upward orientation.

$\frac{713}{180}$
Find the flux of $F$ across $S$ if $F (x, y, z) = ⟨x, - z, y⟩$ and $S$ is the part of the sphere $x^{2} + y^{2} + z^{2} = 4$ in the first octant, oriented towards the origin.

$- \frac{4}{3} π$
Find the flux of $F$ across $S$ if $F (x, y, z) = ⟨0, y, - z⟩$ and $S$ is a positively oriented surface which consists of the paraboloid $y = x^{2} + z^{2}$ , $0 \leq y \leq 1$ , and the disk $x^{2} + z^{2} \leq 1$ , $y = 1$ .

$0$
Find the centroid of the hemisphere $x^{2} + y^{2} + z^{2} = a^{2}$ , $z \geq 0$ .

$(0, 0, a / 2)$
A fluid has density $870$ $kg / m^{3}$ and flows with velocity $v = z i + y^{2} j + x^{2} k$ , where $x$ , $y$ , and $z$ are measured in meters and the components of $v$ are measured in meters per second. Find the rate of flow outward through the cylinder $x^{2} + y^{2} = 4$ , $0 \leq z \leq 1$ .

$0$ $\frac{kg}{s}$