Question

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

Answer #1

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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.

Let H be the hemisphere x2 + y2 + z2 = 54, z ≥ 0, and suppose f
is a continuous function with f(2, 5, 5) = 9, f(2, −5, 5) = 11,
f(−2, 5, 5) = 12, and f(−2, −5, 5) = 13. By dividing H into four
patches, estimate the value below. (Round your answer to the
nearest whole number.)
H
f(x, y, z) dS

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.

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

Use spherical coordinates. Evaluate (2 − x2 − y2) dV, where H is
the solid hemisphere x2 + y2 + z2 ≤ 25, z ≥ 0. H

Use spherical coordinates.
Evaluate
(2 − x2 − y2) dV, where H is
the solid hemisphere x2 + y2 + z2
≤ 25, z ≥ 0.
H

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=

Suppose f(x,y,z)=x2+y2+z2f(x,y,z)=x2+y2+z2 and WW is the solid
cylinder with height 55 and base radius 44 that is centered about
the z-axis with its base at z=−1z=−1. Enter θ as
theta.
with limits of integration
A = 0
B = 2pi
C = 0
D = 4
E = -1
F = 4
(b). Evaluate the integral

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

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