Use green’s theorem to evaluate the integral i = z c 2y 3 dx − 2x 3 dy where c is the circle of radius 2 centered at the origin, oriented counterclockwise

-48π

Step-by-step explanation:

Recall that Green's theorem states the following:

Let P, Q be -functions on a region A, which is the interior of a closed piecewise -path C, parametrized counterclockwise. Then  

We have

Therefore

where A is the interior of the circle of radius 2 centered at the origin.

If we change to polar coordinates

x = rcos(t)

y = rsin(t)

Then A can be described with the inequalities

0 < r < 2

0 < t < 2π  

and

geometric series converge the common ratio r has magnitude less than one.

which is equivalent to

for the example we need

|2-x| < 1

-1 < 2-x   and 2-x < 1

x < 3 and x > 1

we can write that

answer: 1 < x < 3

1a. common ratio r=250/25=10 > 1     diverges

1b.   r=50/100=1/2 converges

when these converge they converge to

where a is the first term.

answer: 200

1c. r=-1   so it's not the case that |r| < 1.     diverges

2a. x is the common ratio so we converge when |x|< 1

2b.   r=-10x / -2 = 5x.

we converge when

|5x| < 1

|x| < 1/5

2c.   r=3-2x

-1 < 3-2x and 3-2x < 1

-4 < -2x and 2 < 2x

x < 2 and 1 < x

1 < x < 2

2d. [/tex] r=-48x^3/12 = 4x^3[/tex]

|4x^3| < 1

|x^3| < \frac 1 4

|x| < 4^{-1/3}

2 1/2 feet

step-by-step explanation: