Let $$P = \left[ {\matrix{ 3 & 1 \cr 1 & 3 \cr } } \right].$$ Consider the set $$S$$ of all vectors $$\left( {\matrix{ x \cr y \cr } } \right)$$ such that $${a^2} + {b^2} = 1$$ where $$\left( {\matrix{ a \cr b \cr } } \right) = P\left( {\matrix{ x \cr y \cr } } \right).$$ Then $$S$$ is

The value of the integral $$\,\,2\int_{ - \infty }^\infty {\left( {{{\sin \,2\pi t} \over {\pi t}}} \right)} dt\,\,$$ is equal to

The value of line integral $$\,\,\int {\left( {2x{y^2}dx + 2{x^2}ydy + dz} \right)\,\,} $$ along a path joining the origin $$(0, 0, 0)$$ and the point $$(1, 1, 1)$$ is

The line integral of the vector field $$\,\,F = 5xz\widehat i + \left( {3{x^2} + 2y} \right)\widehat j + {x^2}z\widehat k\,\,$$ along a path from $$(0, 0, 0)$$ to $$(1,1,1)$$ parameterized by $$\left( {t,{t^2},t} \right)$$ is _________.