Two times the difference between seven times x and four

Use the divergence theorem to calculate the surface integral s f · ds; that is, calculate the flux of f across s. f(x, y, z) = x

valkas [14]

$\mathbf f(x,y,z)=x^4\,\mathbf i-x^3z^2\,\mathbf j+4xy^2z\,\mathbf k$
$\mathrm{div}(\mathbf f)=\dfrac{\partial(x^4)}{\partial x}+\dfrac{\partial(-x^3z^2)}{\partial y}+\dfrac{\partial(4xy^2z)}{\partial z}=4x^3+0+4xy^2=4x(x^2+y^2)$

Let $\mathcal D$ be the region whose boundary is $\mathcal S$ . Then by the divergence theorem,

$\displaystyle\iint_{\mathcal S}\mathbf f\cdot\mathrm d\mathbf S=\iiint_{\mathcal D}4x(x^2+y^2)\,\mathrm dV$

Convert to cylindrical coordinates, setting

$x=r\cos\theta$
$y=r\sin\theta$

and keeping $z$ as is. Then the volume element becomes

$\mathrm dV=r\,\mathrm dr\,\mathrm d\theta\,\mathrm dz$

and the integral is

$\displaystyle\iiint_{\mathcal D}4x(x^2+y^2)\,\mathrm dV=\int_{\theta=0}^{\theta=2\pi}\int_{r=0}^{r=1}\int_{z=0}^{z=r\cos\theta+7}4r\cos\theta\cdot r^2\cdot r\,\mathrm dz\,\mathrm dr\,\mathrm d\theta$
$=\displaystyle4\iiint_{\mathcal D}r^4\cos\theta\,\mathrm dz\,\mathrm dr\,\mathrm d\theta$
$=\dfrac{2\pi}3$

4 0

4 years ago

Adrian jogs 3/4 mile each morning how many days will take to jog 3 miles

lbvjy [14]

----------------------------------------------------------------
Given Information
----------------------------------------------------------------
Adrian can run 3/4 mile in 1 morning.

----------------------------------------------------------------
Find how long he needs to run 1 mile
----------------------------------------------------------------
3/4 miles = 1 morning

[ Divide by 3/4 on both side ]
3/4 ÷ 3/4 miles = 1 ÷ 3/4

1 miles = 4/3 morning

----------------------------------------------------------------
Find how long he needs to take to run 3 miles
----------------------------------------------------------------
1 miles = 4/3 morning

[ multiply by 3 through ]
3 miles = 4/3 x 3
3 miles = 4 mornings

----------------------------------------------------------------
Answer : 4 mornings
----------------------------------------------------------------

6 0

3 years ago

To graph the equation 2x + 5y = 10, Zeplyn draws a line through the points (5, 0) and (0, 2). What is the slope of the line repr

BARSIC [14]

Hi there! The slope of the line is -2/5.

We can determine the slope of a line by dividing the change of y-coordinates by the change of x-coordinates.

In this situation we have a...
change of x-coordinates of (5 - 0) = 5
change of y-coordinates of (0 - 2) = -2

Therefore, the slope of this line is -2/5

6 0

4 years ago

Jessica picked oranges for 2 days during school vacation. She was paid $0.45 per basket for the first 100 baskets picked each da

Mama L [17]

Answer:

$91.75

Step-by-step explanation:

Jessica is paid: