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
mafiozo [28]
3 years ago
5

Use the divergence theorem to calculate the surface integral s f · ds; that is, calculate the flux of f across s. f(x, y, z) = x

4i − x3z2j + 4xy2zk, s is the surface of the solid bounded by the cylinder x2 + y2 = 1 and the planes z = x + 7 and z = 0.
Mathematics
1 answer:
valkas [14]3 years ago
4 0
\mathbf f(x,y,z)=x^4\,\mathbf i-x^3z^2\,\mathbf j+4xy^2z\,\mathbf k
\mathrm{div}(\mathbf f)=\dfrac{\partial(x^4)}{\partial x}+\dfrac{\partial(-x^3z^2)}{\partial y}+\dfrac{\partial(4xy^2z)}{\partial z}=4x^3+0+4xy^2=4x(x^2+y^2)


Let \mathcal D be the region whose boundary is \mathcal S. Then by the divergence theorem,

\displaystyle\iint_{\mathcal S}\mathbf f\cdot\mathrm d\mathbf S=\iiint_{\mathcal D}4x(x^2+y^2)\,\mathrm dV

Convert to cylindrical coordinates, setting

x=r\cos\theta
y=r\sin\theta

and keeping z as is. Then the volume element becomes


\mathrm dV=r\,\mathrm dr\,\mathrm d\theta\,\mathrm dz

and the integral is

\displaystyle\iiint_{\mathcal D}4x(x^2+y^2)\,\mathrm dV=\int_{\theta=0}^{\theta=2\pi}\int_{r=0}^{r=1}\int_{z=0}^{z=r\cos\theta+7}4r\cos\theta\cdot r^2\cdot r\,\mathrm dz\,\mathrm dr\,\mathrm d\theta
=\displaystyle4\iiint_{\mathcal D}r^4\cos\theta\,\mathrm dz\,\mathrm dr\,\mathrm d\theta
=\dfrac{2\pi}3
You might be interested in
How do I write 0.35 as money
Rina8888 [55]
You can right it as 35 cents
8 0
3 years ago
Read 2 more answers
Solve: 5/12 × 8/15=​
Cerrena [4.2K]

Answer:0.5

Step-by-step explanation: If you flip the numbers and the continue to multiply them across and then divide them after you should get 0.5

8 0
3 years ago
Claire has received scores of​ 85, 88,​ 87, and 80 on her algebra tests. what is the minimum score she must receive on the fifth
algol [13]

85+88+87+80+x/5=82

340+x=410

410-340=70

5 0
3 years ago
WHAT IS TRUE ABOUT THE HEIGHT OF THE RECTANGLE
scZoUnD [109]

Answer:

It is totaly based on the diagram

6 0
2 years ago
Read 2 more answers
What is two-thirds a number plus 4 is 7
andriy [413]
2/3x + 4 = 7
         -4    -4
4 cancels out.
2/3x = 3
x 3      x3
6/3 = 2x = 9
         /2     /2
2 cancels out.
x = 9/2 = 4.5

3 0
3 years ago
Read 2 more answers
Other questions:
  • I need help I need the answer and the explanation
    5·1 answer
  • Which point represents the value of -(-2) on the number line
    7·2 answers
  • Please explain
    6·1 answer
  • If you answer these trick questions I will give you brainliest!<br><br> 2+2=<br> 3+3=<br> 1+1=
    12·2 answers
  • Please Help!!! I will give extra points and mark u brainiest!!!!!
    9·2 answers
  • Which table represents a linear function?
    10·2 answers
  • I need it today please help me
    10·2 answers
  • 3) determine the value of <br> (-6) x (-3)
    9·1 answer
  • Pls help meeeee :D show ur work
    9·2 answers
  • Shonda has $125. She spends 35% of the money she has on a concert ticket. How much did Shonda pay for the concert ticket?
    14·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!