For the vector field G = (yeay + 4 cos(4x + y))i + (xeªy + cos(4x + y))) , find the line integral of G along the curve C from the origin along the x-axis to the point (4,0) and then counterclockwise around the circumference of the circle x² + y² = 16 to the point (4/√2, 4/√√2).
For the vector field G = (yeay + 4 cos(4x + y))i + (xeªy + cos(4x + y))) , find the line integral of G along the curve C from the origin along the x-axis to the point (4,0) and then counterclockwise around the circumference of the circle x² + y² = 16 to the point (4/√2, 4/√√2).
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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![### Line Integral of a Vector Field
Consider the vector field
\[
\vec{G} = (y e^{xy} + 4 \cos(4x + y))\hat{i} + (x e^{xy} + \cos(4x + y))\hat{j}
\]
**Objective**:
Calculate the line integral of \(\vec{G}\) along the curve \(C\). The path \(C\) consists of two segments:
1. A line along the \(x\)-axis from the origin to the point \((4, 0)\).
2. A circular path counterclockwise along the circumference of the circle defined by \(x^2 + y^2 = 16\), reaching the point \((\frac{4}{\sqrt{2}}, \frac{4}{\sqrt{2}})\).
**Integral Setup**:
\[
\int_{C} \vec{G} \cdot d\vec{r} = \boxed{}
\]
Complete the calculation to determine the line integral value for the vector field \(\vec{G}\) over the specified path.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff35f8f89-65fc-4b90-98e9-68f8eaa1b3da%2Fdc87ba19-b2f6-456a-bee6-d34144a5aec2%2Fnpyo1qi_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Line Integral of a Vector Field
Consider the vector field
\[
\vec{G} = (y e^{xy} + 4 \cos(4x + y))\hat{i} + (x e^{xy} + \cos(4x + y))\hat{j}
\]
**Objective**:
Calculate the line integral of \(\vec{G}\) along the curve \(C\). The path \(C\) consists of two segments:
1. A line along the \(x\)-axis from the origin to the point \((4, 0)\).
2. A circular path counterclockwise along the circumference of the circle defined by \(x^2 + y^2 = 16\), reaching the point \((\frac{4}{\sqrt{2}}, \frac{4}{\sqrt{2}})\).
**Integral Setup**:
\[
\int_{C} \vec{G} \cdot d\vec{r} = \boxed{}
\]
Complete the calculation to determine the line integral value for the vector field \(\vec{G}\) over the specified path.
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