Question

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

Answer #1

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

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.

Find the minimum of f(x, y, z) = x2 + y2 +
z2 subject to the two constraints x + y + z = 1 and 4x +
5y + 6z = 10

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

show that the function f(x,y,z) =
1/√(x2+y2+z2) provides the
equation fxx + fyy + fzz = 0, called the 3−D Laplace equation.

Find the minimum of f(x,y,z) = x2 + y2 +
z2 subject to the two constraints x + 2y + z = 3 and x -
y = 4 by answering following questions
a) write out the lagrange equation involving lagrange
multipliers λ(lamba) and μ(mu)
b) solve for lamba in terms of x and y
c) solve for x,y,z using the constraints
d) determine the minimum value

Find the linear approximation of the function f(x, y, z) = x2 +
y2 + z2 at (6, 2, 9) and use it to approximate the number 6.012 +
1.972 + 8.982 . (Round your answer to five decimal places.) f(6.01,
1.97, 8.98) ≈

Given the function f(x, y, z) = (x2 + y2 +
z2 )−1/2
a) what is the gradient at the point (12,0,16)?
b) what is the directional derivative of f in the direction of
the vector u = (1,1,1) at the point (12,0,16)?

Use Lagrange multipliers to find the extremal values of
f(x,y,z)=2x+2y+z subject to the
constraint
x2+y2+z2=9.

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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