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Find ((x + 4y)i + 4y j) · dr where C consists of the three line segments from (3,0, 0) to (3, 5,0) to (0,5, 0) to (0, 5, 3). L((x? + 4v)i + 4v3 ). dr = 3248

Find ((x + 4y)i + 4y j) · dr where C consists of the three line segments from (3,0, 0) to (3, 5,0) to (0,5, 0) to (0, 5, 3). L((x? + 4v)i + 4v3 ). dr = 3248

Calculus: Early Transcendentals
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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I got 3248 but that was incorrect. How do I solve this?

**Problem Statement:** Find \(\int_{C} ((x^2 + 4y) \vec{i} + 4y^3 \vec{j}) \cdot d\vec{r}\) where \(C\) consists of the three line segments from \((3, 0, 0)\) to \((3, 5, 0)\) to \((0, 5, 0)\) to \((0, 5, 3)\). **Solution:** \[ \int_{C} ((x^2 + 4y) \vec{i} + 4y^3 \vec{j}) \cdot d\vec{r} = 3248 \] **Explanation of Terms and Concepts:** 1. **Vector Field:** The expression \((x^2 + 4y) \vec{i} + 4y^3 \vec{j}\) represents a vector field. 2. **Line Integral:** The problem asks for the line integral of the vector field along the path \(C\). 3. **Curve \(C\):** Defined by three line segments: - From \((3, 0, 0)\) to \((3, 5, 0)\) - From \((3, 5, 0)\) to \((0, 5, 0)\) - From \((0, 5, 0)\) to \((0, 5, 3)\) This calculation involves parameterizing each segment, evaluating the vector field along the parameterized curve, and computing the dot product with \(d\vec{r}\). The overall solution for this line integral is 3248, which represents the cumulative effect of the field along the specified path.
Transcribed Image Text:**Problem Statement:** Find \(\int_{C} ((x^2 + 4y) \vec{i} + 4y^3 \vec{j}) \cdot d\vec{r}\) where \(C\) consists of the three line segments from \((3, 0, 0)\) to \((3, 5, 0)\) to \((0, 5, 0)\) to \((0, 5, 3)\). **Solution:** \[ \int_{C} ((x^2 + 4y) \vec{i} + 4y^3 \vec{j}) \cdot d\vec{r} = 3248 \] **Explanation of Terms and Concepts:** 1. **Vector Field:** The expression \((x^2 + 4y) \vec{i} + 4y^3 \vec{j}\) represents a vector field. 2. **Line Integral:** The problem asks for the line integral of the vector field along the path \(C\). 3. **Curve \(C\):** Defined by three line segments: - From \((3, 0, 0)\) to \((3, 5, 0)\) - From \((3, 5, 0)\) to \((0, 5, 0)\) - From \((0, 5, 0)\) to \((0, 5, 3)\) This calculation involves parameterizing each segment, evaluating the vector field along the parameterized curve, and computing the dot product with \(d\vec{r}\). The overall solution for this line integral is 3248, which represents the cumulative effect of the field along the specified path.
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