Consider the two-dimensional vector field $$\overrightarrow F (x,y) - x\overrightarrow i + y\overrightarrow j $$, where $$\overrightarrow i $$ and $$\widehat j$$ denote the unit vectors along the x-axis and the y-axis, respectively. A contour C in the x-y plane, as shown in the figure, is composed of two horizontal lines connected at the two ends by two semicircular arcs of unit radius. The contour is traversed in the counter-clockwise sense. The value of the closed path integral
$$\oint\limits_C {\overrightarrow F (x,y)\,.\,(dx\overrightarrow i + dy\overrightarrow j )} $$
is ___________.
The function f(x) = 8loge x $$-$$ x2 + 3 attains its minimum over the interval [1, e] at x = __________.
(Here loge x is the natural logarithm of x.)
The value of the integral
$$\int\!\!\!\int\limits_D {3({x^2} + {y^2})dx\,dy} $$,
where D is the shaded triangular region shown in the diagram, is ___________ (rounded off to the nearest integer).
$${{dy} \over {dx}} = - {\left( {{x \over y}} \right)^n}$$
for n = –1 and n = 1 respectively, are