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itegral s Rdr y Fram Evaluate the where R2 = xt porial P(3,4 } fa) along the duied path op (b)alang the dinect path of P the path Ofg P tis the aigun. 6) alang P (3,4) 윤(3,0)

itegral s Rdr y Fram Evaluate the where R2 = xt porial P(3,4 } fa) along the duied path op (b)alang the dinect path of P the path Ofg P tis the aigun. 6) alang P (3,4) 윤(3,0)

Introductory Circuit Analysis (13th Edition)
Introductory Circuit Analysis (13th Edition)
13th Edition
ISBN:9780133923605
Author:Robert L. Boylestad
Publisher:Robert L. Boylestad
Chapter1: Introduction
Section: Chapter Questions
Problem 1P: Visit your local library (at school or home) and describe the extent to which it provides literature...
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U can use the second picture to derive part b and c I know how to do part a can u explain clearly how to do part b and c I am lost on the parameter of the integration and how to setup the integration operation ?
**Example A (continued 3/4)** (b) along the path \( OP_1P \) \[ C = OP_1P = OP_1 + P_1P \] \[ d\vec{\ell} = d\vec{y}; \quad [OP_1] \] \[ d\vec{\ell} = d\vec{x}; \quad [P_1P] \] \[ \int_C r^2 \, d\vec{\ell} = \int_0^{P_1} r^2 \, d\vec{y} + \int_{P_1}^P r^2 \, d\vec{x} \] \[ = \hat{a_y} \int_0^1 (0 + y^2) \, dy + \hat{a_x} \int_0^1 (x^2 + 1) \, dx \] \[ = \hat{a_y} (1/3) + \hat{a_x} (4/3) \] **Diagram Explanation:** The diagram is a graph with points labeled on a coordinate plane. The path being evaluated is highlighted as follows: - Point \( O \) is at the origin \((0,0)\). - Point \( P_1 \) is at \( (0, 1) \). - Point \( P_2 \) is at \( (1, 0) \). - Point \( P \) is at \( (1, 1) \). The path \( OP_1P \) is shown with lines indicating movement first along the \( y \)-axis from \( O \) to \( P_1 \), and then along the \( x \)-axis from \( P_1 \) to \( P \). The calculations are shown for integrating along these paths separately.
Transcribed Image Text:**Example A (continued 3/4)** (b) along the path \( OP_1P \) \[ C = OP_1P = OP_1 + P_1P \] \[ d\vec{\ell} = d\vec{y}; \quad [OP_1] \] \[ d\vec{\ell} = d\vec{x}; \quad [P_1P] \] \[ \int_C r^2 \, d\vec{\ell} = \int_0^{P_1} r^2 \, d\vec{y} + \int_{P_1}^P r^2 \, d\vec{x} \] \[ = \hat{a_y} \int_0^1 (0 + y^2) \, dy + \hat{a_x} \int_0^1 (x^2 + 1) \, dx \] \[ = \hat{a_y} (1/3) + \hat{a_x} (4/3) \] **Diagram Explanation:** The diagram is a graph with points labeled on a coordinate plane. The path being evaluated is highlighted as follows: - Point \( O \) is at the origin \((0,0)\). - Point \( P_1 \) is at \( (0, 1) \). - Point \( P_2 \) is at \( (1, 0) \). - Point \( P \) is at \( (1, 1) \). The path \( OP_1P \) is shown with lines indicating movement first along the \( y \)-axis from \( O \) to \( P_1 \), and then along the \( x \)-axis from \( P_1 \) to \( P \). The calculations are shown for integrating along these paths separately.
**Title: Evaluating the Integral of \( R^2 \) Along Different Paths** **Objective:** Evaluate the integral \(\int R^2 \, dl\) where \(R^2 = x^2 + y^2\), from the origin \(O\) to the point \(P(3,4)\). **Paths for Evaluation:** 1. **Path (a):** Along the direct path \(OP\) 2. **Path (b):** Along the direct path \(OAP\) 3. **Path (c):** Along the path \(OP_2P\) **Diagram Explanation:** - **Coordinates:** - Origin \(O(0,0)\) - Point \(A(0,4)\) - Point \(P_2(3,0)\) - Point \(P(3,4)\) The diagram illustrates three paths: - **Path \(OP\):** A diagonal line directly from \(O\) to \(P(3,4)\). - **Path \(OAP\):** A two-segment path, first going vertically to \(A(0,4)\) and then horizontally to \(P(3,4)\). - **Path \(OP_2P\):** A two-segment path, first going horizontally to \(P_2(3,0)\) and then vertically to \(P(3,4)\). **Approach:** This exercise requires evaluating an integral along different paths between defined points, exploring how the integral of \(R^2\) varies based on the path taken. This helps illustrate properties such as path independence in vector fields when applicable.
Transcribed Image Text:**Title: Evaluating the Integral of \( R^2 \) Along Different Paths** **Objective:** Evaluate the integral \(\int R^2 \, dl\) where \(R^2 = x^2 + y^2\), from the origin \(O\) to the point \(P(3,4)\). **Paths for Evaluation:** 1. **Path (a):** Along the direct path \(OP\) 2. **Path (b):** Along the direct path \(OAP\) 3. **Path (c):** Along the path \(OP_2P\) **Diagram Explanation:** - **Coordinates:** - Origin \(O(0,0)\) - Point \(A(0,4)\) - Point \(P_2(3,0)\) - Point \(P(3,4)\) The diagram illustrates three paths: - **Path \(OP\):** A diagonal line directly from \(O\) to \(P(3,4)\). - **Path \(OAP\):** A two-segment path, first going vertically to \(A(0,4)\) and then horizontally to \(P(3,4)\). - **Path \(OP_2P\):** A two-segment path, first going horizontally to \(P_2(3,0)\) and then vertically to \(P(3,4)\). **Approach:** This exercise requires evaluating an integral along different paths between defined points, exploring how the integral of \(R^2\) varies based on the path taken. This helps illustrate properties such as path independence in vector fields when applicable.
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