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

Sketch the curves and identify the region bounded by
$$x = {1 \over 2},x = 2,y = \ln \,x$$ and $$y = {2^x}.$$ Find the area of this region.

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 $${y^2} + \int\limits_0^x {f\left( t \right)dt = 2!} $$

Compute the area of the region bounded by the curves $$\,y = ex\,\ln x$$ and $$y = {{\ln x} \over {ex}}$$ where $$ln$$ $$e=1.$$