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puteri [66]
3 years ago
7

Use Stokes' Theorem to evaluate C F · dr where C is oriented counterclockwise as viewed from above. F(x, y, z) = 2yi + xzj + (x

+ y)k, C is the curve of intersection of the plane z = y + 6 and the cylinder x2 + y2 = 1.
Mathematics
1 answer:
Lady_Fox [76]3 years ago
4 0

By Stokes' theorem, the line integral of \vec F over C is given by the surface integral of the curl of \vec F over S, where S is the region of intersection of the plane z=y+6 and the cylinder x^2+y^2=1 with S having positive/upward orientation.

Parameterize S by

\vec s(u,v)=u\cos v\,\vec\imath+u\sin v\,\vec\jmath+(u\sin v+6)\,\vec k

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

Take the normal vector to S to be

\vec s_u\times\vec s_v=-u\,\vec\jmath+u\,\vec k

The curl of \vec F is

\nabla\times\vec F(x,y,z)=(1-x)\,\vec\imath-\vec\jmath+(z-2)\,\vec k

Then the line integral is equivalent to

\displaystyle\int_C\vec F\cdot\mathrm d\vec r=\iint_S(\nabla\times\vec F)\cdot\mathrm d\vec S

=\displaystyle\int_0^{2\pi}\int_0^1\bigg((1-u\cos v)\,\vec\imath-\vec\jmath+(u\sin v+4)\,\vec k\bigg)\cdot\bigg(-u\,\vec\jmath+u\,\vec k\bigg)\,\mathrm du\,\mathrm dv

=\displaystyle\int_0^{2\pi}\int_0^1(5u+u^2\sin v)\,\mathrm du\,\mathrm dv=\boxed{5\pi}

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two arcs of concentric circles are intercepted by the same central angle. the resulting arc length of the arc of the larger circ
Bond [772]
the complete question in the attached figure

Let
A°-----------> <span>the same central angle

</span><span>For the larger circle
length of larger circle=2*pi*r-------> 2*pi*12-----> 75.36 m

360</span>°<span> (full circle) has a length of------------> 75.36 m 
 A</span>°-----------------------------------> 16 m
A°=(16)*360/75.36--------> A=76.43°

For the smaller circle
length of smaller circle=2pi*r-------> 2*pi*7.5-----> 47.1 m

360° (full circle) has a length of------------> 47.1 m 
 76.43°-----------------------------------> X
X=76.43*47.1/360-----> X=10 m ----> length arc of the smaller circle

the answer is
the length arc of the smaller circle is 10 m

3 0
3 years ago
What is 4/7÷2/3? please help me with this​
SIZIF [17.4K]

Answer:

Hi! The correct answer is 8/21!

Step-by-step explanation:

<em><u>~Simplify the expression~</u></em>

4 0
3 years ago
Read 2 more answers
A car rental costs $50 for the first day. Additional days cost $35 per day, unless the car is rented for 7 days or more, in whic
son4ous [18]

Answer:

\$50+\$31.5x

Step-by-step explanation:

Let

x------> the number of days

y----> the cost of renting a car

we know that

For x

y=\$50+\$35x

For x\geq 7\ days

The rate is equal to

0.90*\$35=\$31.5

so

y=\$50+\$31.5x

In this problem. the car has been rented for more than a week

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x> 7\ days

The cost of renting a car is equal to

y=\$50+\$31.5x

6 0
3 years ago
Growing up, Joe lived in a small town. When he left for college, the population was 840.
bearhunter [10]

Answer:882

Step-by-step explanation:5 percent of 840 is 42 so 840 plus 42

8 0
3 years ago
I need help understanding how to solve this problem.
LUCKY_DIMON [66]

Answer:

  • <u><em>D. (-5/13, -12/13)</em></u>

Explanation:

It is indicated to <em>find the coordinates</em> of the <em>point T (5/13, 12/13) </em>at a distance <em>π + x.</em>

  • π is an arc of half a circle, i.e. 180º.

Then, the distance π+ x means that the <em>point T</em> is rotated 180º.

The rule for the rotation of a point 180º around the origin is:

  • (x, y) → (-x, - y)

Applying that rule to the point T:

  • (5/13, 12/13) → (-5/13, - 12/13) ← answer
6 0
3 years ago
Read 2 more answers
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