Let A1 be the area of the region bounded by the curves y = sinx, y = cosx and y-axis in th…

Question

Let A1 be the area of the region bounded by the curves y = sinx, y = cosx and y-axis in the first quadrant. Also, let A2 be the area of the region bounded by the curves y = sinx, y = cosx, x-axis and x = $${\pi  \over 2}$$ in the first quadrant. Then,

Accepted Answer

$${A_1} + {A_2} = \int\limits_0^{\pi /2} {\cos x.\,dx = \left. {\sin x} \right|_0^{\pi /2}}  = 1$$

$${A_1} = \left. {\int\limits_0^{\pi /4} {(\cos x - \sin x)dx = (\sin x + \cos x)} } \right|_0^{\pi /4} = \sqrt 2  - 1$$

$$ \therefore $$ $${A_2} = 1 - \left( {\sqrt 2  - 1} \right) = 2 - \sqrt 2 $$

$$ \therefore $$ $${{{A_1}} \over {{A_2}}} = {{\sqrt 2  - 1} \over {\sqrt 2 \left( {\sqrt 2  - 1} \right)}} = {1 \over {\sqrt 2 }}$$

Solutions · 1