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2 years ago

Determine whether the equation is linear or not. Then graph the equation by finding and plotting ordered-pair

Mathematics

1 answer:

Ghella [55]2 years ago

4 0

Answer:

it is linear the above given equation.

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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

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

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3 years ago

Kent worked in the housewares section of a department store. This year he set a record high for vacuum sales with 567 vacuums so

Mashcka [7]

It is a 5% increase from last year

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2 years ago

What is 20 min ealy than 12:00

Alekssandra [29.7K]

Answer:

11:40

Step-by-step explanation:

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2 years ago

Read 2 more answers

In a scale model of a television, 1 centimeter represents 5 inches.

Thepotemich [5.8K]

Answer:

a)8cm

b)30inches

Step-by-step explanation:

5 0

2 years ago

Find the value of X, Z, and Y in the graph below?

ella [17]

9514 1404 393

Answer:

(x, y, z) = (49, 58, 122)

Step-by-step explanation:

The same-side exterior angles have a sum of 180°.

(x +9) +(3x -25) = 180

4x -16 = 180

x -4 = 45

x = 49

Vertical angles are congruent:

y = 49 +9 = 58

z = 180 -y = 122 = 3(49) -25

The values of x, y, and z are ...

X = 49
Z = 122
Y = 58

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2 years ago