The problem provided is as follows:Let $\vec{F}(x, y) = (x+1) \vec{i}$, and let $C$ be the portion of the parabola $y = x^2$ that starts at $(0, 0)$ and ends at $(2, 4)$.(a) Without doing any calculations, state whether $\int_{C} \vec{F} \cdot d\vec{r}$ is positive, negative, or zero. Clearly state your answer, and explain based on the diagram to the right how you know your answer is correct.**Explanation of the diagram:**The diagram shows the curve $C$, which is part of the parabola $y = x^2$, ranging from $(0, 0)$ to $(2, 4)$ on the xy-plane. The curve is plotted against a field of vectors (depicted as arrows) pointing upward. These vectors, $\vec{F}(x, y) = (x+1) \vec{i}$, are aligned along the positive x-direction and increase in magnitude as x increases.Since the vectors are always horizontal (along the x-axis) and the curve progresses from left to right while increasing in both x and y, the dot product $\vec{F} \cdot d\vec{r}$ will be positive. This is because the direction of $\vec{F}$ and the tangent vector along the curve are in the same general direction, resulting in a positive dot product.Therefore, $\int_{C} \vec{F} \cdot d\vec{r}$ is positive. The alignment of the vectors with the direction of movement along the curve confirms this conclusion visually.

Answered: Image /qna-images/answer/6dd79a70-e6fa-45f5-881a-2ef0af6f7e62.jpg

Let F(r, y) = (x + 1)7, and let C be the portion of the parabola y = x² that starts at (0,0) and ends at (2, 4). (a) Without doing any calculations, state whether fF•dř° is positive, neg- ative, or zero. Clearly state your answer, and explain based on the diagram to the right how you know your answer is correct.

Let F(r, y) = (x + 1)7, and let C be the portion of the parabola y = x² that starts at (0,0) and ends at (2, 4). (a) Without doing any calculations, state whether fF•dř° is positive, neg- ative, or zero. Clearly state your answer, and explain based on the diagram to the right how you know your answer is correct.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

The problem provided is as follows:

Let $\vec{F}(x, y) = (x+1) \vec{i}$, and let $C$ be the portion of the parabola $y = x^2$ that starts at $(0, 0)$ and ends at $(2, 4)$.

(a) Without doing any calculations, state whether $\int_{C} \vec{F} \cdot d\vec{r}$ is positive, negative, or zero. Clearly state your answer, and explain based on the diagram to the right how you know your answer is correct.

**Explanation of the diagram:**

The diagram shows the curve $C$, which is part of the parabola $y = x^2$, ranging from $(0, 0)$ to $(2, 4)$ on the xy-plane. The curve is plotted against a field of vectors (depicted as arrows) pointing upward. These vectors, $\vec{F}(x, y) = (x+1) \vec{i}$, are aligned along the positive x-direction and increase in magnitude as x increases.

Since the vectors are always horizontal (along the x-axis) and the curve progresses from left to right while increasing in both x and y, the dot product $\vec{F} \cdot d\vec{r}$ will be positive. This is because the direction of $\vec{F}$ and the tangent vector along the curve are in the same general direction, resulting in a positive dot product.

Therefore, $\int_{C} \vec{F} \cdot d\vec{r}$ is positive. The alignment of the vectors with the direction of movement along the curve confirms this conclusion visually.

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