Let $${g_i}:\left[ {{\pi \over 8},{{3\pi } \over 8}} \right] \to R,i = 1,2$$, and $$f:\left[ {{\pi \over 8},{{3\pi } \over 8}} \right] \to R$$ be functions such that $${g_1}(x) = 1,{g_2}(x) = |4x - \pi |$$ and $$f(x) = {\sin ^2}x$$, for all $$x \in \left[ {{\pi \over 8},{{3\pi } \over 8}} \right]$$. Define $${S_i} = \int\limits_{{\pi \over 8}}^{{{3\pi } \over 8}} {f(x).{g_i}(x)dx} $$, i = 1, 2

The value of $${{16{S_1}} \over \pi }$$ is _____________.

Let $${g_i}:\left[ {{\pi \over 8},{{3\pi } \over 8}} \right] \to R,i = 1,2$$, and $$f:\left[ {{\pi \over 8},{{3\pi } \over 8}} \right] \to R$$ be functions such that $${g_1}(x) = 1,{g_2}(x) = |4x - \pi |$$ and $$f(x) = {\sin ^2}x$$, for all $$x \in \left[ {{\pi \over 8},{{3\pi } \over 8}} \right]$$. Define $${S_i} = \int\limits_{{\pi \over 8}}^{{{3\pi } \over 8}} {f(x).{g_i}(x)dx} $$, i = 1, 2

The value of $${{48{S_2}} \over {{\pi ^2}}}$$ is ___________.

Let $$M = \{ (x,y) \in R \times R:{x^2} + {y^2} \le {r^2}\} $$, where r > 0. Consider the geometric progression $${a_n} = {1 \over {{2^{n - 1}}}}$$, n = 1, 2, 3, ...... . Let S₀ = 0 and for n $$\ge$$ 1, let S_n denote the sum of the first n terms of this progression. For n $$\ge$$ 1, let C_n denote the circle with center (S_n$$-$$1, 0) and radius a_n, and D_n denote the circle with center (S_n$$-$$1, S_n$$-$$1) and radius a_n.

Consider M with $$r = {{1025} \over {513}}$$. Let k be the number of all those circles C_n that are inside M. Let l be the maximum possible number of circles among these k circles such that no two circles intersect. Then

Let $$M = \{ (x,y) \in R \times R:{x^2} + {y^2} \le {r^2}\} $$, where r > 0. Consider the geometric progression $${a_n} = {1 \over {{2^{n - 1}}}}$$, n = 1, 2, 3, ...... . Let S₀ = 0 and for n $$\ge$$ 1, let S_n denote the sum of the first n terms of this progression. For n $$\ge$$ 1, let C_n denote the circle with center (S_n$$-$$1, 0) and radius a_n, and D_n denote the circle with center (S_n$$-$$1, S_n$$-$$1) and radius a_n.

Consider M with $$r = {{({2^{199}} - 1)\sqrt 2 } \over {{2^{198}}}}$$. The number of all those circles D_n that are inside M is