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1. Parts (a) (b) concern a line integral fe F·dr through the vector field F(x, y) = (1 – x) j. For each given curve C, do the following: (i) Without doing any calculations, determine whether f F · dr is positive or negative, and explain your answer. (ii) Verify your answer to part (a) by calculating F.dr. (b) C: F(t) = (-1+ 4t) i + (3 – 4t) j, 0

1. Parts (a) (b) concern a line integral fe F·dr through the vector field F(x, y) = (1 – x) j. For each given curve C, do the following: (i) Without doing any calculations, determine whether f F · dr is positive or negative, and explain your answer. (ii) Verify your answer to part (a) by calculating F.dr. (b) C: F(t) = (-1+ 4t) i + (3 – 4t) j, 0

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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**Line Integral Assignment** 1. Parts (a)–(b) concern a line integral \(\int_C \vec{F} \cdot d\vec{r}\) through the vector field \(\vec{F}(x, y) = (1-x) \vec{j}\). For each given curve \(C\), do the following: (i) Without doing any calculations, determine whether \(\int_C \vec{F} \cdot d\vec{r}\) is positive or negative, and explain your answer. (ii) Verify your answer to part (a) by calculating \(\int_C \vec{F} \cdot d\vec{r}\). (a) **Curve**: \(C: \vec{r}(t) = \sin t \, \vec{i} + \cos t \, \vec{j}\), \(0 \leq t \leq 2\pi\) - **Diagram Explanation**: The diagram shows a circle centered at the origin \((0,0)\) in the \(xy\)-plane. Arrows representing the vector field point in the negative \(y\)-direction, with magnitude decreasing as \(x\) increases. (b) **Curve**: \(C: \vec{r}(t) = (-1 + 4t) \, \vec{i} + (3 - 4t) \, \vec{j}\), \(0 \leq t \leq 1\) - **Diagram Explanation**: The diagram shows a straight path from \((-1, 3)\) to \((3, -1)\) in the \(xy\)-plane. The vector field arrows point downwards, with their length decreasing as \(x\) increases.
Transcribed Image Text:**Line Integral Assignment** 1. Parts (a)–(b) concern a line integral \(\int_C \vec{F} \cdot d\vec{r}\) through the vector field \(\vec{F}(x, y) = (1-x) \vec{j}\). For each given curve \(C\), do the following: (i) Without doing any calculations, determine whether \(\int_C \vec{F} \cdot d\vec{r}\) is positive or negative, and explain your answer. (ii) Verify your answer to part (a) by calculating \(\int_C \vec{F} \cdot d\vec{r}\). (a) **Curve**: \(C: \vec{r}(t) = \sin t \, \vec{i} + \cos t \, \vec{j}\), \(0 \leq t \leq 2\pi\) - **Diagram Explanation**: The diagram shows a circle centered at the origin \((0,0)\) in the \(xy\)-plane. Arrows representing the vector field point in the negative \(y\)-direction, with magnitude decreasing as \(x\) increases. (b) **Curve**: \(C: \vec{r}(t) = (-1 + 4t) \, \vec{i} + (3 - 4t) \, \vec{j}\), \(0 \leq t \leq 1\) - **Diagram Explanation**: The diagram shows a straight path from \((-1, 3)\) to \((3, -1)\) in the \(xy\)-plane. The vector field arrows point downwards, with their length decreasing as \(x\) increases.
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