Consider a square with vertices at $$(1,1), (-1,1), (-1,-1)$$ and $$(1, -1)$$. Let $$S$$ be the region consisting of all points inside the square which are nearer to the origin than to any edge. Sketch the region $$S$$ and find its area.

In what ratio does the $$x$$-axis divide the area of the region
bounded by the parabolas $$y = 4x - {x^2}$$ and $$y = {x^2} - x?$$

Sketch the region bounded by the curves $$y = {x^2}$$ and
$$y = {2 \over {1 + {x^2}}}.$$ Find the area.

If $$'f$$ is a continuous function with $$\int\limits_0^x {f\left( t \right)dt \to \infty } $$ as $$\left| x \right| \to \infty ,$$ then show that every line $$y=mx$$
intersects the curve $${y^2} + \int\limits_0^x {f\left( t \right)dt = 2!} $$