Let C Be the Circle in Space with the Parameterization $\vec { r } ( t ) = \cos t \vec { i } + \sin t \vec { k } , \quad 0 \leq t \leq 2 \pi$

question 12

Essay

Let C be the circle in space with the parameterization $\vec { r } ( t ) = \cos t \vec { i } + \sin t \vec { k } , \quad 0 \leq t \leq 2 \pi$ Evaluate $\int_{C} \vec{F} \cdot d \vec{r}$ where $\vec { F } = 2 z e ^ { y ^ { 2 } } \vec { i } + \sin \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right) \vec { j } + x \left( y ^ { 2 } - 1 \right) \vec { k }$

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Definitions:

Stamp Act

A 1765 British law that imposed a tax on all paper documents in the American colonies, sparking widespread protest and helping to fuel the American Revolutionary movement.

Declare Independence

The act of a region or group formally stating their intention to become independent and govern themselves, often through a formal declaration.

British Militaries

The armed forces of the United Kingdom, consisting of the Royal Navy, British Army, and Royal Air Force.

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The first President of the United States, serving from 1789 to 1797, a Founding Father, and Commander-in-Chief of the Continental Army during the American Revolutionary War.