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. **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.

Answered: Image /qna-images/answer/bf66b3be-c7d4-4d5e-a513-8ae1842e7dfd.jpg

Answered: itegral s Rdr y Fram Evaluate the where…

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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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Question

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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.

**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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