2. Evaluate each of the following line integrals in two ways*. F (x, y) = (2x – cos y) i + (x sin y) j and – co 3 | F- dr, where | (a) C1 is the straight-line path from (–4,0) to (0,5). F<(r, y) = (2x – y)i+ (2x + y) j and C2 is a circle of radius 4 centered at the origin, traversed once counterclockwise starting at (4,0). (b) | F2· dr, where Acceptable ways to evaluate the integral: • Directly: Parametrize the path and write the integral in terms of your parametrization. • Using the Fundamental Theorem of Line Integrals: Write the theorem and show what you substitute for each part. • Using Green's Theorem: Write the theorem and show what you substitute for each part.

2. Evaluate each of the following line integrals in two ways*. F (x, y) = (2x – cos y) i + (x sin y) j and – co 3 | F- dr, where | (a) C1 is the straight-line path from (–4,0) to (0,5). F<(r, y) = (2x – y)i+ (2x + y) j and C2 is a circle of radius 4 centered at the origin, traversed once counterclockwise starting at (4,0). (b) | F2· dr, where Acceptable ways to evaluate the integral: • Directly: Parametrize the path and write the integral in terms of your parametrization. • Using the Fundamental Theorem of Line Integrals: Write the theorem and show what you substitute for each part. • Using Green's Theorem: Write the theorem and show what you substitute for each part.

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

2. Evaluate each of the following line integrals in two ways*.
F (x, y) = (2x – cos y) i+ (x sin y) and
C1 is the straight-line path from (-4,0) to (0,5).
(a)
| F dr, where
C1
F-(x, y) =
(2x – y) i + (2x + y) j and
C2 is a circle of radius 4 centered at the origin,
traversed once counterclockwise starting at (4,0).
(b) / F2· dĩ, where
F2 · dr, where
C2
Acceptable ways to evaluate the integral:
*
• Directly:
Parametrize the path and write the integral in terms of your parametrization.
• Using the Fundamental Theorem of Line Integrals:
Write the theorem and show what you substitute for each part.
• Using Green's Theorem:
Write the theorem and show what you substitute for each part.

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