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
hram777 [196]
3 years ago
14

Evaluate the surface integral. s x2 + y2 + z2 ds s is the part of the cylinder x2 + y2 = 4 that lies between the planes z = 0 an

d z = 3, together with its top and bottom disks
Mathematics
1 answer:
Leya [2.2K]3 years ago
7 0
Parameterize the lateral face T_1 of the cylinder by

\mathbf r_1(u,v)=(x(u,v),y(u,v),z(u,v))=(2\cos u,2\sin u,v

where 0\le u\le2\pi and 0\le v\le3, and parameterize the disks T_2,T_3 as

\mathbf r_2(r,\theta)=(x(r,\theta),y(r,\theta),z(r,\theta))=(r\cos\theta,r\sin\theta,0)
\mathbf r_3(r,\theta)=(r\cos\theta,r\sin\theta,3)

where 0\le r\le2 and 0\le\theta\le2\pi.

The integral along the surface of the cylinder (with outward/positive orientation) is then

\displaystyle\iint_S(x^2+y^2+z^2)\,\mathrm dS=\left\{\iint_{T_1}+\iint_{T_2}+\iint_{T_3}\right\}(x^2+y^2+z^2)\,\mathrm dS
=\displaystyle\int_{u=0}^{u=2\pi}\int_{v=0}^{v=3}((2\cos u)^2+(2\sin u)^2+v^2)\left\|{{\mathbf r}_1}_u\times{{\mathbf r}_2}_v\right\|\,\mathrm dv\,\mathrm du+\int_{r=0}^{r=2}\int_{\theta=0}^{\theta=2\pi}((r\cos\theta)^2+(r\sin\theta)^2+0^2)\left\|{{\mathbf r}_2}_r\times{{\mathbf r}_2}_\theta\right\|\,\mathrm d\theta\,\mathrm dr+\int_{r=0}^{r=2}\int_{\theta=0}^{\theta=2\pi}((r\cos\theta)^2+(r\sin\theta)^2+3^2)\left\|{{\mathbf r}_3}_r\times{{\mathbf r}_3}_\theta\right\|\,\mathrm d\theta\,\mathrm dr
=\displaystyle2\int_{u=0}^{u=2\pi}\int_{v=0}^{v=3}(v^2+4)\,\mathrm dv\,\mathrm du+\int_{r=0}^{r=2}\int_{\theta=0}^{\theta=2\pi}r^3\,\mathrm d\theta\,\mathrm dr+\int_{r=0}^{r=2}\int_{\theta=0}^{\theta=2\pi}r(r^2+9)\,\mathrm d\theta\,\mathrm dr
=\displaystyle4\pi\int_{v=0}^{v=3}(v^2+4)\,\mathrm dv+2\pi\int_{r=0}^{r=2}r^3\,\mathrm dr+2\pi\int_{r=0}^{r=2}r(r^2+9)\,\mathrm dr
=136\pi
You might be interested in
Please explain how you got your answers! (: Solve the problem by using the counting principle.
Natasha2012 [34]
A. The number of fabric-pattern-color combinations is 4 * 13 * 9 = 468

B. P(1st choice) = no of novels / total books = 3/6 = 1/2
    P(2nd choice) = no of remaining novels/ total remaining books = 2/5
    P(both novels) = 1/2 * 2/5 = 1/5          (without replacement assumed)

C. P(1st choice) = no of biographies / total books = 2/6 = 1/3
    P(2nd choice) = no of remaining biographies/ total remaining books =        1/5
    P(both biographies) = 1/3 * 1/5 = 1/15          (without replacement                assumed)

D. P(1st choice) = no of history books / total books = 1/6
    P(2nd choice) = no of novels/ total remaining books = 3/5
    P(a history, then a novel) = 1/6 * 3/5 = 1/10          (without replacement assumed)
7 0
3 years ago
Please help me my homework is due tomorrow use the screen shot
Ad libitum [116K]

Answer:

1/5

Step-by-step explanation:

(5^7*5^5)/5^13 (when multiplying numbers with the same base, add the exponents)

= (5^12)/5^13 (when dividing exponents with the same base, subtract the exponents)

= 1/5^1 (anything to the first power is itself)

= 1/5

6 0
3 years ago
Nikki has $1100 and spends $35 per week. Mich has $520 and saves $55 per week. In about how many weeks, will Nikki and Micah hav
Rashid [163]

Answer:

6 weeks

Step-by-step explanation:

Given that :

Nikki :

Initial amount = $1100 ;

Amount spent per week = $35

Mich:

Initial amount = $520

Amount saved per week = $55

Number of weeks they'll both have the same amount :

Let number of weeks = w

For Nikki who is spending :

1100 - 35w - - - (1)

For Mich who is saving :

520 + 55w - - - (2)

Equate (1) and (2)

1100 - 35w = 520 + 55w

1100 - 520 = 55w + 35w

580 = 90w

w = 580 / 90

w = 6.444

About 6 weeks

5 0
2 years ago
Help me please it’s not a or b btw
Serga [27]

Answer:

from what I'm seeing the answer is C

5 0
3 years ago
Read 2 more answers
0.3d - 12 = 0.6<br> what’s the value of d?
Wittaler [7]

Answer:

d= 42

Step-by-step explanation:


3 0
3 years ago
Read 2 more answers
Other questions:
  • ASAP! GIVING BRAINLIEST! Please read the question THEN answer correctly! No guessing. Show your work or give an explaination.
    11·1 answer
  • Need help ASAP math
    7·1 answer
  • -4(X - 5) is equivalent to 4-5 + x)<br> False<br> True
    14·2 answers
  • Erica travels 4 blocks right and 4 blocks up between 2 locations on the map. Which two locations does she travel between? (4 poi
    10·2 answers
  • PLEASE HELP!!! thank you :)
    5·2 answers
  • Which linear inequality is represented by the graph?<br> PLEAe help
    15·1 answer
  • 5 pts! (Yall I really need help) How much does a video game cost per in:
    10·2 answers
  • What is 2x2x2x2x2x4x0
    15·2 answers
  • Which pair of angles are alternate interior angles? ​
    9·1 answer
  • PLEASE HELP MEE i dont really understand this<br><br> 4(p+1.25)=7 2/5
    10·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!