. C is the part of the circle x2 + y? = 36 in the upper half plane y > 0, oriented counter-clockwise. Evaluate Sc(2+ y) dx + (4x + ev*) dy. Warning: C is not a closed curve.
. C is the part of the circle x2 + y? = 36 in the upper half plane y > 0, oriented counter-clockwise. Evaluate Sc(2+ y) dx + (4x + ev*) dy. Warning: C is not a closed curve.
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![**Section II2:**
**Problem Statement:**
*C is the part of the circle defined by the equation \(x^2 + y^2 = 36\) that lies in the upper half-plane (\(y \geq 0\)), oriented counter-clockwise. Evaluate the integral:*
\[
\int_C (2 + y) \, dx + (4x + e^{y^2}) \, dy.
\]
**Important Note:**
*Warning: The curve \(C\) is not a closed curve.*
**Explanation:**
The problem involves evaluating a line integral along a semicircular path \(C\), which represents the upper half of a circle with radius 6 (since \(x^2 + y^2 = 36\)).
The integral to evaluate is:
\[
\int_C (2 + y) \, dx + (4x + e^{y^2}) \, dy
\]
This is a line integral of a vector field, where the vector field is given by \((2 + y, 4x + e^{y^2})\).
Since \(C\) is not a closed curve, special care is needed to consider boundary conditions or additional contributions if extending to a closed path.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F67429e19-8d15-47bc-bbfe-dfa923849540%2Fb710e3a9-37ea-4455-a6c7-82b93180db9b%2F93weg9f_processed.png&w=3840&q=75)
Transcribed Image Text:**Section II2:**
**Problem Statement:**
*C is the part of the circle defined by the equation \(x^2 + y^2 = 36\) that lies in the upper half-plane (\(y \geq 0\)), oriented counter-clockwise. Evaluate the integral:*
\[
\int_C (2 + y) \, dx + (4x + e^{y^2}) \, dy.
\]
**Important Note:**
*Warning: The curve \(C\) is not a closed curve.*
**Explanation:**
The problem involves evaluating a line integral along a semicircular path \(C\), which represents the upper half of a circle with radius 6 (since \(x^2 + y^2 = 36\)).
The integral to evaluate is:
\[
\int_C (2 + y) \, dx + (4x + e^{y^2}) \, dy
\]
This is a line integral of a vector field, where the vector field is given by \((2 + y, 4x + e^{y^2})\).
Since \(C\) is not a closed curve, special care is needed to consider boundary conditions or additional contributions if extending to a closed path.
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