1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
tekilochka [14]
3 years ago
12

Evaluate the surface integral:S

Mathematics
1 answer:
rjkz [21]3 years ago
4 0
Assuming S does not include the plane z=0, we can parameterize the region in spherical coordinates using

\mathbf r(u,v)=\left\langle3\cos u\sin v,3\sin u\sin v,3\cos v\right\rangle

where 0\le u\le2\pi and 0\le v\le\dfrac\pi/2. We then have

x^2+y^2=9\cos^2u\sin^2v+9\sin^2u\sin^2v=9\sin^2v
(x^2+y^2)=9\sin^2v(3\cos v)=27\sin^2v\cos v

Then the surface integral is equivalent to

\displaystyle\iint_S(x^2+y^2)z\,\mathrm dS=27\int_{u=0}^{u=2\pi}\int_{v=0}^{v=\pi/2}\sin^2v\cos v\left\|\frac{\partial\mathbf r(u,v)}{\partial u}\times \frac{\partial\mathbf r(u,v)}{\partial u}\right\|\,\mathrm dv\,\mathrm du

We have

\dfrac{\partial\mathbf r(u,v)}{\partial u}=\langle-3\sin u\sin v,3\cos u\sin v,0\rangle
\dfrac{\partial\mathbf r(u,v)}{\partial v}=\langle3\cos u\cos v,3\sin u\cos v,-3\sin v\rangle
\implies\dfrac{\partial\mathbf r(u,v)}{\partial u}\times\dfrac{\partial\mathbf r(u,v)}{\partial v}=\langle-9\cos u\sin^2v,-9\sin u\sin^2v,-9\cos v\sin v\rangle
\implies\left\|\dfrac{\partial\mathbf r(u,v)}{\partial u}\times\dfrac{\partial\mathbf r(u,v)}{\partial v}\|=9\sin v

So the surface integral is equivalent to

\displaystyle243\int_{u=0}^{u=2\pi}\int_{v=0}^{v=\pi/2}\sin^3v\cos v\,\mathrm dv\,\mathrm du
=\displaystyle486\pi\int_{v=0}^{v=\pi/2}\sin^3v\cos v\,\mathrm dv
=\displaystyle486\pi\int_{w=0}^{w=1}w^3\,\mathrm dw

where w=\sin v\implies\mathrm dw=\cos v\,\mathrm dv.

=\dfrac{243}2\pi w^4\bigg|_{w=0}^{w=1}
=\dfrac{243}2\pi
You might be interested in
The tens digit is 9 more than the ones digit my number is?
yKpoI14uk [10]
90 or 100 something like that.
5 0
3 years ago
Please help me with this! I will rate you brainliest!!
lesya [120]

do 6x8.(theres two x's and two y's)

3 0
3 years ago
Read 2 more answers
These points are linear.<br> Find the slope.
Harlamova29_29 [7]

Answer:

here

Step-by-step explanation:

8/1 or 8

8 0
3 years ago
Find BC. Round to the nearest tenth
melamori03 [73]
Use the law of sines; see the attached picture.

6 0
3 years ago
Read 2 more answers
Use the table to determine whether the relationship is proportional. If so, write an equation for the relationship. Tell what ea
Rudik [331]

Answer:

3 is not 4 is proportional

Step-by-step explanation:

8 0
2 years ago
Other questions:
  • Solve for x: (5 points) (-6/5)x -4 = -46<br><br> a. 23 <br> b. 35 <br> c. −23 <br> d. −35
    8·1 answer
  • If ABCD is a parallogram what is the value of x​
    8·2 answers
  • 6x^4-17x^3-7x^2+13x=3 what are all the solutions
    11·1 answer
  • Can anyone please help??
    13·1 answer
  • Factor the quadratic expression
    9·1 answer
  • A and B are independent events. P(A)=0.40 and P(B)=0.30. what is P(A and B)?
    7·2 answers
  • can someone please explain what is happening to rectangle A? Answer choice c says: rotate rectangle A 90 counterclockwise about
    10·1 answer
  • HELP The minute hand of a clock is 5 inches long. to the the nearest tenth of an inch, how far does the tip of the minute hand t
    12·1 answer
  • Divide Rs90 in the ratio 1:2​
    8·1 answer
  • Nine less than one-fourth of a<br> number is three.
    6·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!