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Calc 3 13.7

 

<== Calculus 3
Number Question Answer
See the work
1. Evaluate \displaystyle \int\!\!\int_{S} \sqrt{ 1 + x^{2} + y^{2} } \: dS where S is the helicoid: r(u, v) = ucos(v)i + usin(v)j + vk, with 0 &le u &le 4, 0 &le v &le 5&pi

397.935 The work
2. Let S be the part of the plane 4x + 4y + z = 4 which lies in the first octant, oriented upward. Find the flux of the vector field F = 2i + 2j + 3k across the surface S.

9.5 The work
3. A fluid has density 3 kg/m^3 and flows in a velocity field v = -yi + xj + 4zk where x, y , and z are measured in meters and the components of v in meters per second.
Find the rate of flow outward through the sphere x^{2} + y^{2} + z^{2} = 16

3216.99 The work
4. Let M be the closed surface that consists of the hemisphere

M_1:  x^2 + y^2 + z^2 = 1,\quad z \ge 0,
and its base
M_2: x^2 + y^2 \le 1,\quad z = 0\, .

Let E be the electric field defined by E = (2x, 2y, 2z). Find the electric flux across M. Write the integral over the hemisphere using spherical coordinates, and use the outward pointing normal.
\iint_{M_1} \mathbf{E} \cdot d\mathbf{S} = 
    \int_a^b\int_c^d f(	heta, \phi)\,d	heta\,d\phi,
where

a = , b = , c = , d = ,

f(&theta, &phi) =

\iint_{M_1} \mathbf{E} \cdot d\mathbf{S} =

\iint_{M_2} \mathbf{E} \cdot d\mathbf{S} = , so

\iint_{M} \mathbf{E} \cdot d\mathbf{S} = .
a = 0
b = 1.571
c = 0
d = 6.283


f(&theta, &phi) =

12.57
0
12.57
The work
5. question Answer The work
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