This exercise is adapted from a problem from the textbook. Consider the line integral [(x - y) dx + (x + y) dy where is the curve going counterclockwise around the triangle with vertices (0,0), (1,0), and (0, 1). (a) Sketch a graph of the triangle described above. triangle. Indicate the direction of the curve on the (b) Find parameterizations of the three line segments forming the edges of the triangle. Note: This works the same way that it did in 3D with t, except with two components each instead of three. Review Section 12.5 for details. (c) Evaluate the integral above by first expressing the line integral as a single-variable in terms of the parameter t.
This exercise is adapted from a problem from the textbook. Consider the line integral [(x - y) dx + (x + y) dy where is the curve going counterclockwise around the triangle with vertices (0,0), (1,0), and (0, 1). (a) Sketch a graph of the triangle described above. triangle. Indicate the direction of the curve on the (b) Find parameterizations of the three line segments forming the edges of the triangle. Note: This works the same way that it did in 3D with t, except with two components each instead of three. Review Section 12.5 for details. (c) Evaluate the integral above by first expressing the line integral as a single-variable in terms of the parameter t.
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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![This exercise is adapted from a problem from the textbook.
Consider the line integral
\[
\int_{C} (x-y) \, dx + (x+y) \, dy
\]
where \( C \) is the curve going counterclockwise around the triangle with vertices \( (0,0) \), \( (1,0) \), and \( (0,1) \).
(a) Sketch a graph of the triangle described above. Indicate the direction of the curve on the triangle.
(b) Find parameterizations of the three line segments forming the edges of the triangle. *Note: This works the same way that it did in 3D with \( t \), except with two components each instead of three. Review Section 12.5 for details.*
(c) Evaluate the integral above by first expressing the line integral as a single-variable in terms of the parameter \( t \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fee253cd8-99d1-4831-bbdb-07df756ee09b%2F3a5747b5-b3fd-40a1-9f60-045d4ca4c886%2Fb3tb8lr_processed.png&w=3840&q=75)
Transcribed Image Text:This exercise is adapted from a problem from the textbook.
Consider the line integral
\[
\int_{C} (x-y) \, dx + (x+y) \, dy
\]
where \( C \) is the curve going counterclockwise around the triangle with vertices \( (0,0) \), \( (1,0) \), and \( (0,1) \).
(a) Sketch a graph of the triangle described above. Indicate the direction of the curve on the triangle.
(b) Find parameterizations of the three line segments forming the edges of the triangle. *Note: This works the same way that it did in 3D with \( t \), except with two components each instead of three. Review Section 12.5 for details.*
(c) Evaluate the integral above by first expressing the line integral as a single-variable in terms of the parameter \( t \).
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