Name: Area Enclosed by a Curve and a Line: Illustration
Uploaded: 2019-07-19
Duration: 3 min 35 s
Description: How would you find the area enclosed between a parabola and a straight line? Watch the video to see the method as an application of definite integrals.

Question

Find the area of the region enclosed in first quadrant of circle

x^{2} + y^{2} = 4

and straight line

y = x

and axis of

x

.

Accepted Answer

Given circle $x^{2} + y^{2} = 4$ , line $y = x$ .

Finding intersection point of circle and line,

$\Rightarrow x^{2} + y^{2} = 4$

$\Rightarrow x^{2} + x^{2} = 4$

$\Rightarrow 2 x^{2} = 4$

$\Rightarrow x^{2} = 2$

$\Rightarrow x = \pm \sqrt{2}$

$\Rightarrow x = y = \pm \sqrt{2}$

So, intersection points are $(\sqrt{2}, \sqrt{2}) and (- \sqrt{2}, - \sqrt{2})$ .

Circle $x^{2} + y^{2} = 4 \Rightarrow x^{2} + y^{2} = {(2)}^{2}$

Circle intersects the positive $x$ -axis at $(2, 0)$ .

Required area $=$ area of the region $B C O B$ $+$ area of the region $B A C B$

Area of the region $B C O B$ $= \int_{0}^{\sqrt{2}} x d x$

$= {[\frac{x^{2}}{2}]}_{0}^{\sqrt{2}}$

$= [\frac{{(\sqrt{2})}^{2}}{2} - 0] = 1$ square units

Area of the region $B A C B$ $= \int_{\sqrt{2}}^{2} (\sqrt{{(2)}^{2} - x^{2}}) d x$

$= {[\frac{x}{2} \sqrt{4 - x^{2}} + \frac{4}{2} \sin^{- 1} \frac{x}{2}]}_{\sqrt{2}}^{2}$

$= [\frac{2}{2} \sqrt{4 - {(2)}^{2}} + \frac{4}{2} \sin^{- 1} \frac{2}{2} - \frac{\sqrt{2}}{2} \sqrt{4 - {(\sqrt{2})}^{2}} - \frac{4}{2} \sin^{- 1} \frac{\sqrt{2}}{2}]$

$= [0 + 2 \sin^{- 1} (1) - 1 - 2 \sin^{- 1} \frac{1}{\sqrt{2}}]$

$= [2 \times \frac{π}{2} - 1 - 2 (\frac{π}{4})]$

$= [π - 1 - \frac{π}{2}]$

$= \frac{π}{2} - 1$ square units

Hence, required area $= 1 + \frac{π}{2} - 1 = \frac{π}{2}$ square units

Find the area of the region enclosed in first quadrant of circle $x^{2} + y^{2} = 4$ and straight line $y = x$ and axis of $x$ .

Important Questions on Application of Integrals

Learn and Prepare for any exam you want !

Find the area of the region enclosed in first quadrant of circle x2+y2=4 and straight line y=x and axis of x.

Important Questions on Application of Integrals

Learn and Prepare for any exam you want !

Find the area of the region enclosed in first quadrant of circle $x^{2} + y^{2} = 4$ and straight line $y = x$ and axis of $x$ .