Graph a line that contains the point (4 3) and has a slope of 1/2

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

Leya [2.2K]

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

$\displaystyle\iint_S(x^2+y^2+z^2)\,\mathrm dS=\left\{\iint_{T_1}+\iint_{T_2}+\iint_{T_3}\right\}(x^2+y^2+z^2)\,\mathrm dS$
$=\displaystyle\int_{u=0}^{u=2\pi}\int_{v=0}^{v=3}((2\cos u)^2+(2\sin u)^2+v^2)\left\|{{\mathbf r}_1}_u\times{{\mathbf r}_2}_v\right\|\,\mathrm dv\,\mathrm du+\int_{r=0}^{r=2}\int_{\theta=0}^{\theta=2\pi}((r\cos\theta)^2+(r\sin\theta)^2+0^2)\left\|{{\mathbf r}_2}_r\times{{\mathbf r}_2}_\theta\right\|\,\mathrm d\theta\,\mathrm dr+\int_{r=0}^{r=2}\int_{\theta=0}^{\theta=2\pi}((r\cos\theta)^2+(r\sin\theta)^2+3^2)\left\|{{\mathbf r}_3}_r\times{{\mathbf r}_3}_\theta\right\|\,\mathrm d\theta\,\mathrm dr$
$=\displaystyle2\int_{u=0}^{u=2\pi}\int_{v=0}^{v=3}(v^2+4)\,\mathrm dv\,\mathrm du+\int_{r=0}^{r=2}\int_{\theta=0}^{\theta=2\pi}r^3\,\mathrm d\theta\,\mathrm dr+\int_{r=0}^{r=2}\int_{\theta=0}^{\theta=2\pi}r(r^2+9)\,\mathrm d\theta\,\mathrm dr$
$=\displaystyle4\pi\int_{v=0}^{v=3}(v^2+4)\,\mathrm dv+2\pi\int_{r=0}^{r=2}r^3\,\mathrm dr+2\pi\int_{r=0}^{r=2}r(r^2+9)\,\mathrm dr$
$=136\pi$

7 0

3 years ago

Two thirds of the students in Harry's math class and three fifths of the students in amanis

prisoha [69]

what's the question your tryna figure out

4 0

3 years ago

Each school day Lauren rides her

Licemer1 [7]

Answer:

21.41667

Step-by-step explanation:

6 2/3 + 4 + 10 3/4 = 21.41667

4 0

3 years ago

Read 2 more answers

If a multiple-choice test consists of 5 questions, each with 4 possible answers of which only 1 is correct, (a) in how many diff

sp2606 [1]

The different ways can a student check off one answer to each question is 1024.

It is required to find the different ways can a student check off one answer to each question.

<h3>What is probability?</h3>

probability is the ratio of the number of favorable outcomes and the total number of possible outcomes. The chance that a particular event (or set of events) will occur expressed on a linear scale from 0 (impossibility) to 1 (certainty), also expressed as a percentage between 0 and 100%.

Given:

If a multiple choice test consists of 5 questions each with 4 possible answers of which only one is correct,

,To find how many different ways can a student check off one answer to each question we have,

: 4^5 = 2^10

= 1024

Therefore, the different ways can a student check off one answer to each question is 1024.

Learn more about probability here:

brainly.com/question/11234923

#SPJ4

5 0

1 year ago

if a population doubles every 5 years how many years would it take for the population to increase by 10 times original amount?

Alecsey [184]

25 years is the answer

7 0

3 years ago

Graph a line that contains the point (4 3) and has a slope of 1/2​

Graph a line that contains the point (4 3) and has a slope of 1/2