The area enclosed by y=sqrt{5-x^{2}} and y=|x | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) The curves are $y=\sqrt{5-x^2}$ (a semicircle of radius $\sqrt{5}$) and $y=|x-1|$.To find the intersection points, set $\sqrt{5-x^2} = |x-1|$.Squa

Vedclass · Accepted Answer

$\left(\frac{5 \pi}{4}-\frac{1}{2}ight) 	ext{ sq units}$

Vedclass · Answer

(C) The curves are $y=\sqrt{5-x^2}$ (a semicircle of radius $\sqrt{5}$) and $y=|x-1|$.To find the intersection points, set $\sqrt{5-x^2} = |x-1|$.Squaring both sides: $5-x^2 = x^2-2x+1 \implies 2x^2-2x-4=0 \implies x^2-x-2=0 \implies (x-2)(x+1)=0$.So, the intersection points are $x=-1$ and $x=2$.The area $A$ is given by $\int_{-1}^{2} (\sqrt{5-x^2} - |x-1|) dx$.$A = \int_{-1}^{2} \sqrt{5-x^2} dx - \int_{-1}^{2} |x-1| dx$.For the first integral, $\int_{-1}^{2} \sqrt{5-x^2} dx = \left[ \frac{x}{2}\sqrt{5-x^2} + \frac{5}{2}\sin^{-1}\left(\frac{x}{\sqrt{5}}\right) \right]_{-1}^{2} = (1 + \frac{5}{2}\sin^{-1}\frac{2}{\sqrt{5}}) - (-1 + \frac{5}{2}\sin^{-1}\frac{-1}{\sqrt{5}}) = 2 + \frac{5}{2}(\sin^{-1}\frac{2}{\sqrt{5}} + \sin^{-1}\frac{1}{\sqrt{5}})$.Since $\sin^{-1}\frac{2}{\sqrt{5}} + \sin^{-1}\frac{1}{\sqrt{5}} = \sin^{-1}(\frac{2}{\sqrt{5}}\sqrt{1-\frac{1}{5}} + \frac{1}{\sqrt{5}}\sqrt{1-\frac{4}{5}}) = \sin^{-1}(1) = \frac{\pi}{2}$, the first integral is $2 + \frac{5\pi}{4}$.For the second integral, $\int_{-1}^{2} |x-1| dx = \int_{-1}^{1} (1-x) dx + \int_{1}^{2} (x-1) dx = [x - \frac{x^2}{2}]_{-1}^{1} + [\frac{x^2}{2} - x]_{1}^{2} = (1 - \frac{1}{2} - (-1 - \frac{1}{2})) + (2 - 2 - (\frac{1}{2} - 1)) = (1.5) + (0.5) = 2$.Thus, $A = 2 + \frac{5\pi}{4} - 2 = \frac{5\pi}{4}$ sq units. Wait, re-evaluating the integral $\int_{-1}^{2} |x-1| dx$: $\int_{-1}^{1} (1-x) dx = [x - x^2/2]_{-1}^{1} = (1-0.5) - (-1-0.5) = 0.5 + 1.5 = 2$. $\int_{1}^{2} (x-1) dx = [x^2/2 - x]_1^2 = (2-2) - (0.5-1) = 0.5$. Total is $2.5 = 5/2$.$A = 2 + \frac{5\pi}{4} - 2.5 = \frac{5\pi}{4} - 0.5$ sq units.

The area enclosed by $y=\sqrt{5-x^{2}}$ and $y=|x-1|$ is

Similar Questions

If the area of the region $\{(x, y): 1+x^2 \leq y \leq \min \{x+7, 11-3x\}\}$ is $A$,then $3A$ is equal to

The area of the region enclosed by the parabola $y=4x-x^2$ and $3y=(x-4)^2$ is equal to

The area (in square units) of the region bounded by the circle $x^2 + y^2 = 9$ and the parabola $y^2 = 8x$ is...

The area bounded by the curves $y = 1 - \cos(\pi x)$,$y = -x^2$ and the lines $x = \frac{1}{2}$ and $x = -\frac{1}{2}$ is:

The area (in sq. units) of the part of the circle $x^2+y^2=169$ which is below the line $5x-y=13$ is $\frac{\pi \alpha}{2 \beta}-\frac{65}{2}+\frac{\alpha}{\beta} \sin ^{-1}\left(\frac{12}{13}\right)$ where $\alpha, \beta$ are coprime numbers. Then $\alpha+\beta$ is equal to . . . . . . .

Vedclass Test Series

Exam Paper Generator

Online Exam Module