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mariarad [96]
3 years ago
10

Helpppppppppppppppppppppppppppppppppppppppppppppppppppppp

Mathematics
1 answer:
muminat3 years ago
5 0
The answer is 17.156
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Use the Divergence Theorem to evaluate S F · dS, where F(x, y, z) = z2xi + y3 3 + sin z j + (x2z + y2)k and S is the top half of
GenaCL600 [577]

Close off the hemisphere S by attaching to it the disk D of radius 3 centered at the origin in the plane z=0. By the divergence theorem, we have

\displaystyle\iint_{S\cup D}\vec F(x,y,z)\cdot\mathrm d\vec S=\iiint_R\mathrm{div}\vec F(x,y,z)\,\mathrm dV

where R is the interior of the joined surfaces S\cup D.

Compute the divergence of \vec F:

\mathrm{div}\vec F(x,y,z)=\dfrac{\partial(xz^2)}{\partial x}+\dfrac{\partial\left(\frac{y^3}3+\sin z\right)}{\partial y}+\dfrac{\partial(x^2z+y^2)}{\partial k}=z^2+y^2+x^2

Compute the integral of the divergence over R. Easily done by converting to cylindrical or spherical coordinates. I'll do the latter:

\begin{cases}x(\rho,\theta,\varphi)=\rho\cos\theta\sin\varphi\\y(\rho,\theta,\varphi)=\rho\sin\theta\sin\varphi\\z(\rho,\theta,\varphi)=\rho\cos\varphi\end{cases}\implies\begin{cases}x^2+y^2+z^2=\rho^2\\\mathrm dV=\rho^2\sin\varphi\,\mathrm d\rho\,\mathrm d\theta\,\mathrm d\varphi\end{cases}

So the volume integral is

\displaystyle\iiint_Rx^2+y^2+z^2\,\mathrm dV=\int_0^{\pi/2}\int_0^{2\pi}\int_0^3\rho^4\sin\varphi\,\mathrm d\rho\,\mathrm d\theta\,\mathrm d\varphi=\frac{486\pi}5

From this we need to subtract the contribution of

\displaystyle\iint_D\vec F(x,y,z)\cdot\mathrm d\vec S

that is, the integral of \vec F over the disk, oriented downward. Since z=0 in D, we have

\vec F(x,y,0)=\dfrac{y^3}3\,\vec\jmath+y^2\,\vec k

Parameterize D by

\vec r(u,v)=u\cos v\,\vec\imath+u\sin v\,\vec\jmath

where 0\le u\le 3 and 0\le v\le2\pi. Take the normal vector to be

\dfrac{\partial\vec r}{\partial v}\times\dfrac{\partial\vec r}{\partial u}=-u\,\vec k

Then taking the dot product of \vec F with the normal vector gives

\vec F(x(u,v),y(u,v),0)\cdot(-u\,\vec k)=-y(u,v)^2u=-u^3\sin^2v

So the contribution of integrating \vec F over D is

\displaystyle\int_0^{2\pi}\int_0^3-u^3\sin^2v\,\mathrm du\,\mathrm dv=-\frac{81\pi}4

and the value of the integral we want is

(integral of divergence of <em>F</em>) - (integral over <em>D</em>) = integral over <em>S</em>

==>  486π/5 - (-81π/4) = 2349π/20

5 0
3 years ago
Help me answer this pls​
stira [4]

Answer:

x=5

Step-by-step explanation:

From this information we know that,

5x - 6 = 3x + 4  \\ 5x - 3x = 4 + 6 \\ 2x = 10 \\ x = 5

Hope it helps :)

6 0
3 years ago
Which of the following number lines shows the solution to the inequality given below? 4x+7&lt;-36 OR 5x+3&gt;-7
MissTica

Answer:

You have the right answer

Step-by-step explanation

4 0
3 years ago
You roll two dice. What is the probability you get a 2 and a number greater than 2?
abruzzese [7]
Hello!

You have to find the probability of getting a 2

This is a 1/6 chance

Then you have to find the chance of getting a number greater than 2

This is a 4/6 chance

To find the chance of these happening at the same time you multiply these together

1/6 * 4/6 = 1/9

There is a 1/9 chance

Hope this helps!
6 0
3 years ago
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Factor completely 2x2 + 2x − 24. 2(x − 3)(x + 4) (2x − 6)(x + 4) (2x − 3)(x + 8) 2(x − 4)(x + 6)
Veseljchak [2.6K]
2<span>x<span><span>​</span></span></span>² <span><span><span>​ ​</span></span></span>+ 2x − 2<span>4

</span><span>GCF = 2

2(2x</span>²/2 + 2x/2 - 24/2)
<span>
2(x - 3)(x + 4)</span>
7 0
3 years ago
Read 2 more answers
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