First slide

Area of bounded Regions

Question

$The area bounded by y = \sin^{- 1} x; y = \cos^{- 1} x, and the x -axis.$

Moderate

A

$\sqrt{2} - 1$
B

$\sqrt{2} + 1$
C

$\frac{\sqrt{2} - 1}{2}$
D

$\frac{\sqrt{2} - 1}{4}$

Solution

$y = \sin^{- 1} x, y = \cos^{- 1} x, and the x -axis is A = \int_{0}^{1 / \sqrt{2}} \sin^{- 1} x d x + \int_{1 / \sqrt{2}}^{1} \cos^{- 1} x d x$
$\begin{aligned} or \\ A = \int_{0}^{π / 4} (\cos y - \sin y) d y \\ \begin{aligned} = [\sin y + \cos y]_{0}^{π / 4} \\ = [\frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}} - 1] \\ = \sqrt{2} - 1 \end{aligned} \end{aligned}$

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