Consider the closed region

bounded simultaneously by the paraboloid and plane, jointly denoted

. By the divergence theorem,

And since we have

the volume integral will be much easier to compute. Converting to cylindrical coordinates, we have




Then the integral over the paraboloid would be the difference of the integral over the total surface and the integral over the disk. Denoting the disk by

, we have

Parameterize

by


which would give a unit normal vector of

. However, the divergence theorem requires that the closed surface

be oriented with outward-pointing normal vectors, which means we should instead use

.
Now,



So, the flux over the paraboloid alone is
Answer:
Pam: $181
Amanda: $362
Julie: $452
Step-by-step explanation:
(What does Mike have to do with this problem?)
Let a = Amanda's pay
Let p = Pam's pay
Let j = Julie's pay
"Amanda made twice what Pam earned"
a = 2p
"Julie made $90 more than Amanda"
j = a + 90
j = 2p + 90
Pam earned p
Total salary
a + p + j = 2p + p + 2p + 90
Total salary
$995
2p + p + 2p + 90 = 995
5p = 905
p = 181
a = 2p = 2(181) = 362
j = 2p + 90 = 362 + 90 = 452
Answer:
Pam: $181
Amanda: $362
Julie: $452
Step-by-step explanation:
csc x / (cot x + tan x)
Write in terms of sine and cosine.
(1 / sin x) / [(cos x / sin x) + (sin x / cos x)]
Multiply top and bottom by sin x.
1 / [cos x + (sin²x / cos x)]
Multiply top and bottom by cos x.
cos x / (cos²x + sin²x)
Use Pythagorean identity.
cos x / 1
cos x