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
Roberto will save 1/6 of his allowance each day. If you get two dollars a day about how much money will you save round to the ne
AlladinOne [14]
About $.33 hint: of means multiply
5 0
3 years ago
Solve −5+1/4x=3x+6 by graphing.
erma4kov [3.2K]

Answer:

hope its right lol

5 0
3 years ago
Vertex A in quadrilateral ABCD lies at (-3, 2). If you rotate ABCD 180° clockwise about the origin, what will be the coordinates
natali 33 [55]

The rule for a rotation by 180° about the origin is (x,y) -->(−x,−y)

So A(-3, 2) to A'(3, -2)

Answer:

A'(3, -2)

4 0
3 years ago
NEED THE ANSWER PLEASE
Alexus [3.1K]

Answer:

d. 89°

Step-by-step explanation:

The given measure of the angles formed are;

m∠AED = 48°, m\widehat{AG} = 175°

According to circle theorem, the angle formed by a chord and a tangent of a circle is given by half of the measure of the arc intercepted by the chord in the direction of the angle;

Therefore;

m∠AED = (1/2) × m\widehat{ABE}  = 48°

∴ m\widehat{ABE} = 2 × 48° = 96°

m∠AEF = (1/2) × m\widehat{AGE}

m\widehat{ABE} + m\widehat{AGE} = 360° (angle round a circle)

∴ m\widehat{AGE} = 360° - m\widehat{ABE} = 360° - 96° = 264°

m\widehat{AGE} = m\widehat{AG} + m\widehat{EG}

∴ m\widehat{EG} = m\widehat{AGE} - m\widehat{AG} = 264° - 175° = 89°

m\widehat{EG} = 89°.

5 0
3 years ago
Use Greens Theorem to evaluate integral x^2ydx - xy^2dy, where C is 0 ≤ y ≤ √9-x^2 with counterclockwise orientation
Ratling [72]

Answer:

Step-by-step explanation:

a circle will satisfy the conditions of Green's Theorem since it is closed and simple.

Let's identify P and Q from the integral

P=x^2 y, and Q= xy^2

Now, using Green's theorem on the line integral gives,

\oint\limits_C {x^2ydx-xy^2dy } =\iint\limits_D {y^2-x^2} \, dA\\\\

3 0
3 years ago
Other questions:
  • Inequalities <br><img src="https://tex.z-dn.net/?f=%20-%205%20%5Ctimes%20%20%2B%202%20%5Cgeqslant%20%20-%2048" id="TexFormula1"
    12·1 answer
  • 6k + 2 = 8 -------------------------------------------------------------someone solve.
    8·1 answer
  • O prove that all circles are similar, you would use similarity
    8·1 answer
  • Round 8.631 to the nearest hundredth
    12·1 answer
  • Whats 4x10^6 in standard form? for brainlist!
    10·2 answers
  • Which relation is not a function?
    11·1 answer
  • Sort each type of expense into the category where it fits best.
    6·1 answer
  • PROBLEM: Riley has a rectangular shaped patio that is 13 feet long by 15 feet wide. He wants to DOUBLE THE AREA of the patio by
    7·1 answer
  • If y=2, then y5 = what
    6·1 answer
  • I met a boy today literally not even an hour ago at a Halloween party and we talked and hung out for like more that half of the
    15·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!