Let $\vec { F } = x ^ { 3 } \vec { i } + \left( x + \sin ^ { 3 } y \right) \vec { j }$

question 7

Essay

Let $\vec { F } = x ^ { 3 } \vec { i } + \left( x + \sin ^ { 3 } y \right) \vec { j }$ (a)Find the line integral $\int _ { C _ { 1 } } \vec { F } \cdot d \vec { r }$ , where C₁ is the line from (0, 0)to ( $\pi$ , 0).
(b)Evaluate the double integral $\int _ { R } 1 d A$ where R is the region enclosed by the curve y = sin x and the x-axis for 0 $\le$ x $\le$ $\pi$ .What is the geometric meaning of this integral?
(c)Use Green's Theorem and the result of part (a)to find $\int _ { c _ { 2 } } \vec { F } \cdot d \vec { r }$ where C₂ is the path from (0, 0)to ( $\pi$ , 0)along the curve y = sin x.

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Definitions:

Biological Difference

Variations in physical and physiological characteristics between individuals or populations, often discussed in the context of gender, race, or species.

Projective Agency

The capacity of individuals to envisage future actions and goals influenced by their desires and aspirations, thereby shaping their future.

Decolonization

The process of deconstructing colonial ideologies, structures, and legacies, and reclaiming and valorizing indigenous cultures and knowledge systems.

Free Human Consciousness

Free human consciousness refers to the human ability to be aware of one's own thoughts and to think independently free from external constraints or influences.