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Fractal20

- 74

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## Homework Statement

Find the value of the line integral

[itex]\int[/itex]

_{C}(e

^{-x^3}- 3y)dx + (tan y + y

^{4}+ x) dy

where C is the counterclockwise-oriented circle of radius 4 centered at (0,2).

## The Attempt at a Solution

First off, I didn't think this was path independent since the derivative of the dx term in respect to y equals -3 which doesn't equal the derivative of the dy term with respect to x, 1.

After getting stuck on some direct approaches, I realized it is probably a Green's Thm application.

This then translates to [itex]\int[/itex][itex]\int[/itex] 4 dA. So in polar coordinates the path of the circle is

x = 4 cos [itex]\vartheta[/itex]

y = 4 sin [itex]\vartheta[/itex] + 2

This is where I get unsure. To try and get the limits on r I tried to say r = sqrt (x + y) and then plugged in the paramatized equations for x and y. This simplified to r = sqrt (20 + 16 sin [itex]\vartheta[/itex]). So then I thought that r varies from 2 to this general thing and hence

[itex]\int[/itex] [itex]^{2pi}_{0}[/itex] [itex]\int[/itex][itex]^{\sqrt{20 + 16 sin\vartheta}_{2}}[/itex] 4 r dr d[itex]\vartheta[/itex].

Is this right? Note that the limits with respect to r is suppose to be 2 to that sqrt expression, the two just kept popping up a little below and to the right... In particular is this a correct way to find the respective limits of integration? Thanks!

## Homework Statement

## Homework Equations

## The Attempt at a Solution

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