The line x=π4 divides the area of the region bounded by y = sinx, y = cos x and x-axis 0≤x≤π2 into two regions of areas A1 and A2. Then A1:A2=

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The line $x = \frac{π}{4}$ divides the area of the region bounded by y = sinx, y = cos x and x-axis $(0 \leq x \leq \frac{π}{2})$ into two regions of areas $A_{1}$ and $A_{2} .$ Then $A_{1} : A_{2} =$

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4 :1

3 :1

2 :1

1 : 1

answer is D.

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Detailed Solution

$y = \sin x, y = \cos x x = x = 0, x = π / 2$

$\begin{aligned} A_{1} = \int_{0}^{\frac{π}{4}} \cos x - \sin x d x \\ = (\sin x + \cos x)_{0}^{\frac{π}{2}} \\ = \sqrt{2} - 1 \end{aligned}$

$\begin{aligned} A_{2} = \int_{\frac{π}{4}}^{π / 2} \sin x - \cos x d x \\ = (- \cos x - \sin x)_{π / 4}^{π / 2} \\ = (- 1) - (\frac{- 1}{\sqrt{2}} - \frac{1}{\sqrt{2}}) \\ = - 1 \sqrt{2} = \sqrt{2} - 1 \end{aligned}$