Which number is NOT in the solution set of the inequality x

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

An evergreen nursery usually sells a certain shrub after 7 years of growth and shaping. The growth rate during those 7 years is

Ne4ueva [31]

Answer:

( 0.6 t^2 + 3t + 11 ) cm

Step-by-step explanation:

dh/dt = 1.2t + 3

at t = 0, h = 11 cm

(a)

dh / dt = 1.2 t + 3

dh = (1.2 t + 3) dt

integrate on both sides

h = 0.6 t^2 + 3t + c .... (1)

where c is the integrating constant

put t = 0

11 = c

Put in equation (1) , we get

h = ( 0.6 t^2 + 3t + 11 ) cm

Thus, teh height of tree after t years is given by

( 0.6 t^2 + 3t + 11 ) cm.

3 0

3 years ago

Hector puts together two tables to make a long table for a class picnic.

lys-0071 [83]

Answer: 14 feet 4 inches

Step-by-step explanation:

Given: The length of the first table = 7 feet 7 inches

The length of the second table = 6 feet 9 inches

The total length of two tables = 7 feet 7 inches + 6 feet 9 inches

= (7+6) feet (7+9) inches

=13 feet 16 inches

Since 1 feet = 12 inches

The total length of two tables = 13 feet+ (12 inches +4 inches)

=13 feet +( 1 feet +4 inches)

= 14 feet 4 inches

Hence, the total length of the two tables = 14 feet 4 inches

7 0

3 years ago

120 < 12k + 29 need help plz answer for 15 points

kherson [118]

1000 this is because it is the right one

8 0

3 years ago

Read 2 more answers

(2,1),(-3,1) how do you solve this I’m looking for the slope I also have to show work

shutvik [7]

Answer: 0

Step-by-step explanation:

To find the slope, you would need to use the formula $m=\frac{y_2-y_1}{x_2-x_1}$ .

$m=\frac{1-1}{-3-2} =\frac{0}{-5} =0$

The slope is 0, meaning the line is horizontal.

7 0

3 years ago

Which number is NOT in the solution set of the inequality x < 15?