Suppose F⃗ (x, y)=4yi⃗ +2xyj⃗ . Use Green's Theorem to calculate the circulation of F⃗ around the perimeter of a circle C of radius 3 centered at the origin and oriented counter-clockwise.

In this exercise it will be necessary to use Green's theorem to calculate the perimeter that will be given by

Knowing that Green's theorem should be used to solve this exercise then:

In this way, you must convert to polar coordinates, we will have:

We can say that the perimeter of the circle is given by:

See more about Green's theorem at link

-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

solution:

as, mean is defined as sum of all the observations divided by total number of observations.

and , median is mid value of the data whether arranged in ascending order or descending order.

consider, roadsters:

2,3,4,5,6,7,8,9

mean =

also, median= , because number of observations is even, so median is mean of two mid values.

now, coming to bandits

2,3,4,5,6

mean =  

median = 4, as number of observations is odd.

so, roadsters has greater median as well mean than bandits.

→option a is true about the repairs performed on 2 types of cars:

bandits have lower median and mean than roadsters.

c.35

step-by-step explanation:

8+13=21

21+8=29

29+6=35