Question

Compute the surface integral over the given oriented surface:

F=〈0,9,x2〉F=〈0,9,x2〉 , hemisphere x2+y2+z2=4x2+y2+z2=4, z≥0z≥0 , outward-pointing normal

Answer #1

Evaluate the surface integral
S
F · dS
for the given vector field F and the oriented
surface S. In other words, find the flux of
F across S. For closed surfaces, use the
positive (outward) orientation.
F(x, y, z) = yi − xj + 2zk,
S is the hemisphere
x2 + y2 + z2 = 4,
z ≥ 0,
oriented downward

Evaluate the surface integral ∫∫S F · dS for the given vector
field F and the oriented surface S. In other words, find the flux
of F across S. For closed surfaces, use the positive (outward)
orientation. F(x, y, z) = xz i + x j + y k S is the hemisphere x2 +
y2 + z2 = 4, y ≥ 0, oriented in the direction of the positive
y-axis. Incorrect: Your answer is incorrect.

Evaluate ∫∫Sf(x,y,z)dS , where f(x,y,z)=0.4sqrt(x2+y2+z2)) and S
is the hemisphere x2+y2+z2=36,z≥0

Compute ∫∫S F·dS for the vector field F(x,y,z)
=〈0,0,x2+y2〉through the surface given by x^2+y^2+z^2= 4, z≥0 with
outward pointing normal.
Please explain and show work.
Thank you so much.

Calculate the line integral of the vector field
?=〈?,?,?2+?2〉F=〈y,x,x2+y2〉 around the boundary curve, the curl of
the vector field, and the surface integral of the curl of the
vector field.
The surface S is the upper hemisphere
?2+?2+?2=36, ?≥0x2+y2+z2=36, z≥0
oriented with an upward‑pointing normal.
(Use symbolic notation and fractions where needed.)
∫?⋅??=∫CF⋅dr=
curl(?)=curl(F)=
∬curl(?)⋅??=∬Scurl(F)⋅dS=

Consider the integral of f(x, y, z) = z + 1 over the upper
hemisphere σ: z = 1 − x2 − y2 (0 ≤ x2 + y2 ≤ 1). (a) Explain why
evaluating this surface integral using (8) results in an improper
integral. (b) Use (8) to evaluate the integral of f over the
surface σr : z = 1 − x2 − y2 (0 ≤ x2 + y2 ≤ r2 < 1). Take the
limit of this result...

Evaluate the surface integral S F · dS for the given vector
field F and the oriented surface S. In other words, find the flux
of F across S. For closed surfaces, use the positive (outward)
orientation. F(x, y, z) = x2 i + y2 j + z2 k S is the boundary of
the solid half-cylinder 0 ≤ z ≤ 25 − y2 , 0 ≤ x ≤ 3

Calculate ∫ ∫S f(x,y,z)dS for the given surface and function.
x2+y2+z2=144, 6≤z≤12; f(x,y,z)=z2(x2+y2+z2)−1.

Evaluate the surface integral S F · dS for the given vector
field F and the oriented surface S. In other words, find the flux
of F across S. For closed surfaces, use the positive (outward)
orientation. F(x, y, z) = yi − xj + 4zk, S is the hemisphere x^2 +
y2^ + z^2 = 4, z ≥ 0, oriented downward

Evaluate the surface integral
S
F · dS
for the given vector field F and the oriented
surface S. In other words, find the flux of
F across S. For closed surfaces, use the
positive (outward) orientation.
F(x, y, z) = x i − z j + y k
S is the part of the sphere
x2 + y2 + z2 = 4
in the first octant, with orientation toward the origin

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