4 Use Green's theorem to evaluate O(1-y') dx+(x* + e ) dy , where C is the boundary of the region between thecircles x + y = 4 and x + y? = 9.+5 X4-3-2-1124-1:

Name: 4 Use Green's theorem to evaluate O(1-y') dx+(x* + e ) dy , where C is
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Description: 4 Use Green's theorem to evaluate O(1-y') dx+(x* + e ) dy , where C is the boundary of the region between thecircles x + y = 4 and x + y? = 9.+5 X4-3-2-1124-1:

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Functions and Change: A Modeling Approach to College Algebra (MindTap Course List)

6th Edition

ISBN:9781337111348

Author:Bruce Crauder, Benny Evans, Alan Noell

Publisher:Bruce Crauder, Benny Evans, Alan Noell

ChapterA: Appendix

SectionA.2: Geometric Constructions

Problem 10P: A soda can has a volume of 25 cubic inches. Let x denote its radius and h its height, both in...

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Use Green's theorem to evaluate O(1-y') dx+(x* + e ) dy , where C is the boundary of the region between the
circles x + y = 4 and x + y? = 9.
+
5 X
4
-3
-2
-1
1
2
4
-1:

Expert Solution

Step 1

The integral is $\oint_{C} (1 - y^{3}) d x + (x^{3} + e^{y^{2}}) d y$ where C is the boundary of the region between the circles $x^{2} + y^{2} = 4$ and $x^{2} + y^{2} = 9$ .

Using Green's theorem, $\oint_{C} P d x + Q d y = \iint_{D} (\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}) d A$ .

Here , $P = 1 - y^{3}$ and $Q = x^{3} + e^{y^{2}}$ .

Then,

$\begin{array}{rcl} \frac{\partial Q}{\partial x} & = & \frac{\partial}{\partial x} (x^{3} + e^{y^{2}}) \\ = & 3 x^{2} \\ \frac{\partial P}{\partial x} & = & \frac{\partial}{\partial x} (1 - y^{3}) \\ = & - 3 y^{2} \end{array}$

Step 2

Sketch the region as shown below.

Evaluate $\oint_{C} (1 - y^{3}) d x + (x^{3} + e^{y^{2}}) d y$ using Green's theorem.

$\oint_{C} (1 - y^{3}) d x + (x^{3} + e^{y^{2}}) d y = \iint_{D} (\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}) d A$

Also,

$\begin{array}{rcl} 4 & \leq & x^{2} + y^{2} \leq 9 \\ 4 & \leq & r^{2} \leq 9 \\ 2 & \leq & r \leq 3 \end{array}$

and $0 \leq θ \leq 2 π$ .

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