Find the line integral of z2 dx + y dy + 4y dz, where C consists of two parts, C, and C,. C, is the intersection of cylinder x2 + y2 = 16 and plane z = 3 from (0, 4, 3) to (-4, 0, 3). C, is a line segment from (-4, 0, 3) to (0, 1, 5). (Round your answer to two decimal places.)
Find the line integral of z2 dx + y dy + 4y dz, where C consists of two parts, C, and C,. C, is the intersection of cylinder x2 + y2 = 16 and plane z = 3 from (0, 4, 3) to (-4, 0, 3). C, is a line segment from (-4, 0, 3) to (0, 1, 5). (Round your answer to two decimal places.)
Calculus: Early Transcendentals
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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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![**Problem Statement: Line Integral Evaluation**
You are asked to find the line integral of the vector field along the curve \( C \), expressed as:
\[ \int_{C} z^2 \, dx + y \, dy + 4y \, dz \]
The curve \( C \) consists of two distinct parts, \( C_1 \) and \( C_2 \):
1. **Curve \( C_1 \)**:
- This is the intersection of a cylinder and a plane.
- The cylinder is defined by the equation \( x^2 + y^2 = 16 \).
- The plane is defined by \( z = 3 \).
- \( C_1 \) spans from the point \((0, 4, 3)\) to the point \((-4, 0, 3)\).
2. **Curve \( C_2 \)**:
- This part of the curve is a straight line segment.
- It starts from the point \((-4, 0, 3)\) and ends at the point \((0, 1, 5)\).
To compute this integral, you need to evaluate the given vector field along both parts of the curve separately and then sum the results. Your final answer should be rounded to two decimal places.
**Additional Details:**
- **Cylinder Equation**: \( x^2 + y^2 = 16 \)
- **Plane Equation**: \( z = 3 \)
- **Curve Segments**:
- From \((0, 4, 3)\) to \((-4, 0, 3)\) for \( C_1 \).
- From \((-4, 0, 3)\) to \((0, 1, 5)\) for \( C_2 \).
Make sure to follow standard techniques for evaluating line integrals, including parameterizing the curves where necessary and applying the appropriate formulas.
Please input your computed answer rounded to two decimal places in the box provided.
\[ \boxed{\hspace{5cm}} \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5fb9488b-2819-46e3-9f11-b4c772cbdaae%2Fea84e910-165a-4066-9e23-5d2cf4bd4405%2Fu7lggb_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement: Line Integral Evaluation**
You are asked to find the line integral of the vector field along the curve \( C \), expressed as:
\[ \int_{C} z^2 \, dx + y \, dy + 4y \, dz \]
The curve \( C \) consists of two distinct parts, \( C_1 \) and \( C_2 \):
1. **Curve \( C_1 \)**:
- This is the intersection of a cylinder and a plane.
- The cylinder is defined by the equation \( x^2 + y^2 = 16 \).
- The plane is defined by \( z = 3 \).
- \( C_1 \) spans from the point \((0, 4, 3)\) to the point \((-4, 0, 3)\).
2. **Curve \( C_2 \)**:
- This part of the curve is a straight line segment.
- It starts from the point \((-4, 0, 3)\) and ends at the point \((0, 1, 5)\).
To compute this integral, you need to evaluate the given vector field along both parts of the curve separately and then sum the results. Your final answer should be rounded to two decimal places.
**Additional Details:**
- **Cylinder Equation**: \( x^2 + y^2 = 16 \)
- **Plane Equation**: \( z = 3 \)
- **Curve Segments**:
- From \((0, 4, 3)\) to \((-4, 0, 3)\) for \( C_1 \).
- From \((-4, 0, 3)\) to \((0, 1, 5)\) for \( C_2 \).
Make sure to follow standard techniques for evaluating line integrals, including parameterizing the curves where necessary and applying the appropriate formulas.
Please input your computed answer rounded to two decimal places in the box provided.
\[ \boxed{\hspace{5cm}} \]
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