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.