Question

The area bounded by the curves $y=\sqrt{5-x^2},y=|x-1|$ is

• A. $\left[\frac{5\mathrm{\pi }}{4}-2\right]$
• B. $\left[\frac{5\mathrm{\pi }-2}{4}\right]$
• C. $\left[\frac{5\mathrm{\pi }-2}{2}\right]$
• D. $\left[\frac{\mathrm{\pi }}{2}-5\right]$

Answer 1

The correct option is B

$\left[\frac{5\mathrm{\pi }-2}{4}\right]$

Explanation for the correct option:

Step 1: Finding intersection point and draw diagram

Given curves are $y=\sqrt{5-x^2},y=|x-1|$

equating both equation and squaring each other we get

$5-x^2={\left(x-1\right)}^2$

$2x^2-2x-4=0$

$x=-1,2$

So these intersection point on $X$ axes are shown in diagram with curve as

Step 2: Forming the integral to find the bounded area as shown

$A={\int }_{-1}^2\sqrt{5-x^2}dx-{\int }_{-1}^1(1-x)dx-{\int }_1^2(x-1)dx$

Step 3: Solving the area integral using integration identities

On integrating

$\begin{array}{rcl}A& =& {\left[\frac{x}{2}\sqrt{5-x^2}+\frac{5}{2}{\sin }^{-1}\left(\frac{x}{\sqrt{5}}\right)\right]}_{-1}^2-{\left[x-\frac{x^2}{2}\right]}^{-1}-{\left[\frac{x^2}{2}-x\right]}_1^2\\ & =& \frac{5}{2}\left({\sin }^{-1}\frac{2}{\sqrt{5}}+{\sin }^{-1}\left(\frac{1}{\sqrt{5}}\right)\right)-\frac{1}{2}\\ & =& \frac{5}{2}{\sin }^{-1}\left(\frac{2}{\sqrt{5}}\sqrt{1-\frac{1}{5}}+\frac{1}{\sqrt{5}}\sqrt{1-\frac{4}{5}}\right)-\frac{1}{2}\\ & =& \frac{5}{2}{\sin }^{-1}\left(1\right)-\frac{1}{2}\\ & =& \frac{5\mathrm{\pi }}{4}-\frac{1}{2}\\ & =& \frac{5\mathrm{\pi }-2}{4}\end{array}$ Using, $$\begin{gathered} {\mathbf{\int }}_{}^{}\sqrt{{\mathbf{a}}^{\mathbf{2}}\mathbf{-}{\mathbf{x}}^{\mathbf{2}}}\mathit{d}\mathit{x}\mathbf{=}\frac{\mathbf{x}}{\mathbf{2}}\sqrt{{\mathbf{a}}^{\mathbf{2}}\mathbf{-}{\mathbf{x}}^{\mathbf{2}}}\mathbf{+}\frac{{\mathbf{a}}^{\mathbf{2}}}{\mathbf{2}}\mathit{s}\mathit{i}{\mathit{n}}^{\mathbf{-}\mathbf{1}}\left(\frac{x}{\sqrt{5}}\right) \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{\int }{\mathit{x}}^{\mathbf{n}}\mathit{d}\mathit{x}\mathbf{=}\frac{{\mathbf{x}}^{\mathbf{n}\mathbf{+}\mathbf{1}}}{\mathbf{n}\mathbf{+}\mathbf{1}} \end{gathered}$$

Hence, the correct option is B.