Write the slope intercept form of the equation 6x + 5y = 32

Let f(x,y,z) = ztan-1(y2) i + z3ln(x2 + 1) j + z k. find the flux of f across the part of the paraboloid x2 + y2 + z = 3 that li

Sophie [7]

Consider the closed region $V$ bounded simultaneously by the paraboloid and plane, jointly denoted $S$ . By the divergence theorem,

$\displaystyle\iint_S\mathbf f(x,y,z)\cdot\mathrm dS=\iiint_V\nabla\cdot\mathbf f(x,y,z)\,\mathrm dV$

And since we have

$\nabla\cdot\mathbf f(x,y,z)=1$

the volume integral will be much easier to compute. Converting to cylindrical coordinates, we have

$\displaystyle\iiint_V\nabla\cdot\mathbf f(x,y,z)\,\mathrm dV=\iiint_V\mathrm dV$
$=\displaystyle\int_{\theta=0}^{\theta=2\pi}\int_{r=0}^{r=1}\int_{z=2}^{z=3-r^2}r\,\mathrm dz\,\mathrm dr\,\mathrm d\theta$
$=\displaystyle2\pi\int_{r=0}^{r=1}r(3-r^2-2)\,\mathrm dr$
$=\dfrac\pi2$

Then the integral over the paraboloid would be the difference of the integral over the total surface and the integral over the disk. Denoting the disk by $D$ , we have

$\displaystyle\iint_{S-D}\mathbf f\cdot\mathrm dS=\frac\pi2-\iint_D\mathbf f\cdot\mathrm dS$

Parameterize $D$ by

$\mathbf s(u,v)=u\cos v\,\mathbf i+u\sin v\,\mathbf j+2\,\mathbf k$
$\implies\mathbf s_u\times\mathbf s_v=u\,\mathbf k$

which would give a unit normal vector of $\mathbf k$ . However, the divergence theorem requires that the closed surface $S$ be oriented with outward-pointing normal vectors, which means we should instead use $\mathbf s_v\times\mathbf s_u=-u\,\mathbf k$ .

Now,

$\displaystyle\iint_D\mathbf f\cdot\mathrm dS=\int_{u=0}^{u=1}\int_{v=0}^{v=2\pi}\mathbf f(x(u,v),y(u,v),z(u,v))\cdot(-u\,\mathbf k)\,\mathrm dv\,\mathrm du$
$=\displaystyle-4\pi\int_{u=0}^{u=1}u\,\mathrm du$
$=-2\pi$

So, the flux over the paraboloid alone is

$\displaystyle\iint_{S-D}\mathbf f\cdot\mathrm dS=\frac\pi2-(-2\pi)=\dfrac{5\pi}2$

6 0

3 years ago

What is the solution to the inequality? 10x + 18 < -2

padilas [110]

Let's solve your inequality step-by-step.

10x+18<−2

Step 1: Subtract 18 from both sides.

10x+18−18<−2−18

10x<−20

Step 2: Divide both sides by 10.

10x/10<−20/10

x<−2

Answer:

x<−2

4 0

3 years ago

The mass of a grain of sand is about 10^-3 gram. About how many grains of sand are in the bag of sand?

nika2105 [10]

there are 10 million grains in that bag of sand this is because one grain equals to 10 power -3 what about 10 kg you change it into grams then you cross multiply and get the answer

8 0

3 years ago

2. How many terms are there in the expression: 3a + 6

Aliun [14]

Answer: 2

Step-by-step explanation: That expression is a binomial in any case the terms in a polynomial are separated by a plus or minus sign (+,-) so you can determine the number of terms by counting how many number or variable are being separated (ex: 3a+a-1 would have 3 terms and be a trinomial)

8 0

3 years ago

I need help with this

Digiron [165]

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6 0

3 years ago

Write the slope intercept form of the equation 6x + 5y = 32​

Write the slope intercept form of the equation 6x + 5y = 32