Let F(x, y) = x²e* ï + 60x²y² j, and let C1 denote the curve consistingof the line segment from (0, 0) to (1,1), followed by the arc of the circlex2 + y? =(1, – 1) to (0,0) (see figure to the right).>(1, 1)2 from (1,1) to (1, –1), followed by the line segment from(0, 0)Calculate Sa. F dr. Show your work, and circleyourfinal answer.* (1, –1)

Answered: Let F(x, y) = x²e*i+ 60x²y² j, and let…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Let F(x, y) = x²e* ï + 60x²y² j, and let C1 denote the curve consisting
of the line segment from (0, 0) to (1,1), followed by the arc of the circle
x2 + y? =
(1, – 1) to (0,0) (see figure to the right).
>(1, 1)
2 from (1,1) to (1, –1), followed by the line segment from
(0, 0)
Calculate Sa. F dr. Show your work, and circle
your
final answer.
* (1, –1)

Expert Solution

Step 1

Given,

$F (x, y) = (x^{2} e^{x}, 60 x^{2} y^{2})$

The curve $C_{1}$ consists of the line segment $C_{a}$ from $(0, 0)$ to $(1, 1)$ , followed by the arc $C_{b}$ of the circle $x^{2} + y^{2} = 2$ from $(1, 1)$ to $(1, - 1)$ , followed by the line segment $C_{c}$ from $(1, - 1)$ to $(0, 0)$

Curve $C_{a} :$ Parametrizing the line segment $C_{a},$

$r_{a} (t) = (t, t), 0 \leq t \leq 1$

Curve $C_{b} :$ Parametrizing the line segment $C_{b},$

$r_{b} (t) = (\sqrt{2} \cos t, \sqrt{2} \sin t), 0 \leq t \leq \frac{3 π}{2}$

Curve $C_{c} :$ Parametrizing the line segment $C_{c},$

$r_{c} (t) = (1 - t, t - 1), 0 \leq t \leq 1$

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