**Problem 1: Evaluate the line integral**\[\int_{C} xy \, dx + x^2y^3 \, dy,\]where $ C $ is the counterclockwise boundary curve around the triangle with vertices $(0, 0)$, $(1, 0)$, and $(1, 2)$, by the two following methods.**Method 1: Calculate the line integral directly.** Let $ C_1 $ be the line segment from $(0, 0)$ to $(1, 0)$, $ C_2 $ be the line segment from $(1, 0)$ to $(1, 2)$, and $ C_3 $ be the line segment from $(1, 2)$ to $(0, 0)$. Then\[\int_{C_1} xy \, dx + x^2y^3 \, dy = 0\]\[\int_{C_2} xy \, dx + x^2y^3 \, dy = \text{(Enter value)}\]\[\int_{C_3} xy \, dx + x^2y^3 \, dy = \text{(Enter value)}\]Therefore,\[\int_{C} xy \, dx + x^2y^3 \, dy = \text{(Enter value)}\]

Given : The integral ∮C xy dx + x2y3 d y , where C is the boundary triangle with vertices (0,0),…

Problem. 1: Evaluate the line integral xy dæ + x²y³ dy, where C is the counterclockwise boundary curve around the triangle with vertices (0, 0), (1,0), and (1, 2), by the two following methods. Method 1: calculate the line integral directly. Let Cı be the line segment from (0, 0) to (1,0), C2 be the line segment from (1,0) to (1, 2), and C3 be the line segment from (1, 2) to (0, 0). Then xy dx + x²y° dy = 0 xy dx + æ²y³ dy = ? xy dx + x²y* dy = C3 Therefore, xy dæ + a²y° dy =

Problem. 1: Evaluate the line integral xy dæ + x²y³ dy, where C is the counterclockwise boundary curve around the triangle with vertices (0, 0), (1,0), and (1, 2), by the two following methods. Method 1: calculate the line integral directly. Let Cı be the line segment from (0, 0) to (1,0), C2 be the line segment from (1,0) to (1, 2), and C3 be the line segment from (1, 2) to (0, 0). Then xy dx + x²y° dy = 0 xy dx + æ²y³ dy = ? xy dx + x²y* dy = C3 Therefore, xy dæ + a²y° dy =

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

**Problem 1: Evaluate the line integral**

\[
\int_{C} xy \, dx + x^2y^3 \, dy,
\]

where $ C $ is the counterclockwise boundary curve around the triangle with vertices $(0, 0)$, $(1, 0)$, and $(1, 2)$, by the two following methods.

**Method 1: Calculate the line integral directly.** Let $ C_1 $ be the line segment from $(0, 0)$ to $(1, 0)$, $ C_2 $ be the line segment from $(1, 0)$ to $(1, 2)$, and $ C_3 $ be the line segment from $(1, 2)$ to $(0, 0)$. Then

\[
\int_{C_1} xy \, dx + x^2y^3 \, dy = 0
\]

\[
\int_{C_2} xy \, dx + x^2y^3 \, dy = \text{(Enter value)}
\]

\[
\int_{C_3} xy \, dx + x^2y^3 \, dy = \text{(Enter value)}
\]

Therefore,

\[
\int_{C} xy \, dx + x^2y^3 \, dy = \text{(Enter value)}
\]

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