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Lunna [17]
3 years ago
11

Suppose F⃗ (x,y)=4yi⃗ +2xyj⃗ . Use Green's Theorem to calculate the circulation of F⃗ around the perimeter of a circle C of radi

us 3 centered at the origin and oriented counter-clockwise.
Mathematics
2 answers:
Andre45 [30]3 years ago
4 0

Answer:

From Green's theorem, the circulation of a function F(x,y) around a circle is given as

∫(F(x,y).dA = Area of the circle

π(3^2) - π(0^2) = 9π

Since the result is oriented counter-clockwise, the result will take negative value.

The circulation of F(x,y) is -9π

Step-by-step explanation:

∫c (4y dx + 2xy dy)

= ∫∫ [(∂/∂x)(2xy) - (∂/∂y)(4y)] dA, by Green's Theorem

By integrating the function F(x,y) = 4yi + 2xyj, around the circle, the result is πr2[3, 0], from origin 0, to radius of 3

Juliette [100K]3 years ago
3 0

Answer:

-9π

Step-by-step explanation:

∫c (4y dx + 2xy dy)

= ∫∫ [(∂/∂x)(2xy) - (∂/∂y)(4y)] dA, by Green's Theorem

= ∫∫ (2y - 4) dA

Now convert to polar coordinates:

∫(r = 0 to 3) ∫(θ = 0 to 2π) (2r sin θ - 4) * (r dθ dr) --- first integration

= ∫(r = 0 to 3) (-2r cos θ - 4θ) * r {for θ = 0 to 2π} dr

= ∫(r = 0 to 3) -2πr dr

= -πr² {for r = 0 to 3}

= -π(3²) - -π(0)²

= -9π

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8 0
2 years ago
Graph the system of equations.<br><br> {8x+8y=642x−2y=−4
klemol [59]
<span>The graph is attached.

Explanation:
We can use the x- and y-intercepts to graph. The x-intercept of the first equation is 8, and the y-intercept is 8. The x-intercept of the second equation is -2, and the y-intercept is 2.
<span>
x-intercepts are where the data crosses the x-axis. At every one of these points, the y-coordinate will be 0; therefore we can substitute 0 for y and solve to get the value of the x-intercept.

For the first equation, we would have
8x+8(0)=64
8x=64.

Divide both sides by 8:
8x/8 = 64/8
x=8.

For the second equation,
2x-2(0)=-4
2x=-4.

Divide both sides by 2:
2x/2 = -4/2
x=-2.

y-intercepts are where the data crosses the y-axis. At every one of these points, the x-coordinate will be 0; therefore we can substitute 0 for x and solve to get the value of the y-intercept.

For the first equation,
8(0)+8y=64
8y=64.

Divide both sides by 8:
8y/8 = 64/8
y=8.

For the second equation,
2(0)-2y=-4
-2y=-4.

Divide both sides by -2:
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6 0
3 years ago
Please hwelp please​
Fantom [35]

Answer:

2⁸ > 2.38 × 10² > 6³ > 2.16 × 10¹ > \sqrt{300}, 300%

Step-by-step explanation:

Given numbers are,

2.38 × 10², √300, 2⁸, 300%, 6³, 2.16 × 10¹

We have to arrange these numbers in the decreasing order(from greatest to least)

2.38 × 10² = 2.38 × 100 = 238

√300 = 10√3 ≈ 17.32

2⁸ = 256

300% = \frac{300}{100} = 3

6³ = 216

2.16 × 10¹ = 21.6

Therefore, decreasing order (from the greatest to least) will be,

256 > 238 > 216 > 21.6 > 17.32 > 3

2⁸ > 2.38 × 10² > 6³ > 2.16 × 10¹ > \sqrt{300}, 300%

5 0
3 years ago
Plz help me get the answer ASAP
Dimas [21]

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8 0
2 years ago
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