Let $$b \ne 0$$ and for $$j=0, 1, 2, ..., n,$$ let $${S_j}$$ be the area of
the region bounded by the $$y$$-axis and the curve $$x{e^{ay}} = \sin $$ by,
$${{jr} \over b} \le y \le {{\left( {j + 1} \right)\pi } \over b}.$$ Show that $${S_0},{S_1},{S_2},\,....,\,{S_n}$$ are in
geometric progression. Also, find their sum for $$a=-1$$ and $$b = \pi .$$

Let $$f(x)$$ be a continuous function given by $$$f\left( x \right) = \left\{ {\matrix{ {2x,} & {\left| x \right| \le 1} \cr {{x^2} + ax + b,} & {\left| x \right| > 1} \cr } } \right\}$$$

Find the area of the region in the third quadrant bounded by the curves $$x = - 2{y^2}$$ and $$y=f(x)$$ lying on the left of the line $$8x+1=0.$$

Let $$f(x)= Maximum $$ $$\,\left\{ {{x^2},{{\left( {1 - x} \right)}^2},2x\left( {1 - x} \right)} \right\},$$ where $$0 \le x \le 1.$$
Determine the area of the region bounded by the curves
$$y = f\left( x \right),$$ $$x$$-axes, $$x=0$$ and $$x=1.$$

Let $${A_n}$$ be the area bounded by the curve $$y = {\left( {\tan x} \right)^n}$$ and the
lines $$x=0,$$ $$y=0,$$ and $$x = {\pi \over 4}.$$ Prove that for $$n > 2,$$
$${A_n} + {A_{n - 2}} = {1 \over {n - 1}}$$ and deduce $${1 \over {2n + 2}} < {A_n} < {1 \over {2n - 2}}.$$