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zloy xaker [14]
3 years ago
5

Evaluate c (y + 7 sin(x)) dx + (z2 + 9 cos(y)) dy + x3 dz where c is the curve r(t) = sin(t), cos(t), sin(2t) , 0 ≤ t ≤ 2π. (hin

t: observe that c lies on the surface z = 2xy.)
Mathematics
1 answer:
saw5 [17]3 years ago
4 0
Treat \mathcal C as the boundary of the region \mathcal S, where \mathcal S is the part of the surface z=2xy bounded by \mathcal C. We write

\displaystyle\int_{\mathcal C}(y+7\sin x)\,\mathrm dx+(z^2+9\cos y)\,\mathrm dy+x^3\,\mathrm dz=\int_{\mathcal C}\mathbf f\cdot\mathrm d\mathbf r

with \mathbf f=(y+7\sin x,z^2+9\cos y,x^3).

By Stoke's theorem, the line integral is equivalent to the surface integral over \mathcal S of the curl of \mathbf f. We have


\nabla\times\mathbf f=(-2z,-3x^2,-1)

so the line integral is equivalent to

\displaystyle\iint_{\mathcal S}\nabla\times\mathbf f\cdot\mathrm d\mathbf S
=\displaystyle\iint_{\mathcal S}\nabla\times\mathbf f\cdot\left(\dfrac{\partial\mathbf s}{\partial u}\times\dfrac{\partial\mathbf s}{\partial v}\right)\,\mathrm du\,\mathrm dv


where \mathbf s(u,v) is a vector-valued function that parameterizes \mathcal S. In this case, we can take

\mathbf s(u,v)=(u\cos v,u\sin v,2u^2\cos v\sin v)=(u\cos v,u\sin v,u^2\sin2v)

with 0\le u\le1 and 0\le v\le2\pi. Then

\mathrm d\mathbf S=\left(\dfrac{\partial\mathbf s}{\partial u}\times\dfrac{\partial\mathbf s}{\partial v}\right)\,\mathrm du\,\mathrm dv=(2u^2\cos v,2u^2\sin v,-u)\,\mathrm du\,\mathrm dv

and the integral becomes

\displaystyle\iint_{\mathcal S}(-2u^2\sin2v,-3u^2\cos^2v,-1)\cdot(2u^2\cos v,2u^2\sin v,-u)\,\mathrm du\,\mathrm dv
=\displaystyle\int_{v=0}^{v=2\pi}\int_{u=0}^{u=1}u-6u^4\sin^3v-4u^4\cos v\sin2v\,\mathrm du\,\mathrm dv=\pi<span />
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Colin &amp; Brian share a lottery win of £2100 in the ratio 4 : 1. Colin then shares his part between himself, his wife &amp; th
Paraphin [41]

Answer:

His wife gets $336 more than his son.

Step-by-step explanation:

Colin and Brian:

Ratio of 4:1.

So Colin gets \frac{4}{4+1} = \frac{4}{5} of the prize.

\frac{4*2100}{5} = 1680

Colin got $1680. Dividing between himself, his wife and their son:

Ratio of 2:5:3.

So Colin gets \frac{2}{2+3+5} = \frac{2}{10}, his wife gets \frac{5}{2+3+5} = \frac{5}{10} and his son gets \frac{3}{2+3+5} = \frac{3}{10}

Wife:

\frac{5*1680}{10} = 840

His wife got $840.

Son:

\frac{3*1680}{10} = 504

His son got $504.

How much more does his wife get over their son?

840 - 504 = 336

His wife gets $336 more than his son.

8 0
2 years ago
Can I get some help please? <br><br><br> Thanks!
quester [9]
An ellipse (oval shape) is expressed by the following equation:

\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2}=1 where h is the x coordinate of the center and k is the y coordinate of the center. Furthermore, a is the horizontal distance from the center, and b is the vertical distance from the center. Lastly, c is the distance from the center to one of the foci (they are spaced apart equally).

We can find the foci by using a^2 - b^2 = c^2

36 - 11 = c^2

c =  \sqrt{25}  = 5

Since the k value in this case is 0, the y value of both foci are 0. Also, since h and k are both 0, we know the center of the ellipse is at the origin.

So the foci are (-5, 0) and (5, 0)

Hope this helps :)

3 0
3 years ago
PLEASE HELP !! ILL GIVE BRAINLIEST *EXTRA 40 POINTS* DONT SKIP :(( .!
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Answer:

D ez

Step-by-step explanation:

6 0
2 years ago
Find the slope of the line that passes through each pair of points (13,-3),(-5,-5)
PIT_PIT [208]

Answer:

\boxed{\bold{\frac{1}{9} }}

Explanation:

(13,-3), (-5,-5)

<u>Find The Slope:</u>

<u />\bold{\mathrm{Slope}=\frac{y_2-y_1}{x_2-x_1}}<u />

<u />\bold{\left(x_1,\:y_1\right)=\left(13,\:-3\right),\:\left(x_2,\:y_2\right)=\left(-5,\:-5\right)}<u />

<u />\bold{m=\frac{-5-\left(-3\right)}{-5-13}}<u />

<u />\bold{ \ m \ = \ \frac{1}{9} }<u />

3 0
3 years ago
The surface area of a ball is 3,600 square millimeters. What is the balls radius
allsm [11]
A = 4 \pi r^2

3600 = 4\pi r^2

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r^2 = 286.479

r = 16.9

Answer: The radius is 16.9 mm.
4 0
3 years ago
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