The area of the region A = { ( x , y ) : 4 x 2 + y 2 8 and y 2 4 x } is: [2026]

Question

The area of the region $A = {(x, y) : 4 x^{2} + y^{2} \leq 8 and y^{2} \leq 4 x}$ is: [2026]

Smart Achievers · Accepted Answer

(3)

$A = \int_{0}^{2} 2 \sqrt{x} d x + 2 \int_{1}^{\sqrt{2}} \sqrt{8 - 4 x^{2}} d x$

$= \frac{8}{3} [x^{3 / 2}] \int_{0}^{1} + 4 \int_{1}^{\sqrt{2}} \sqrt{2 - x^{2}} d x$

$= \frac{8}{3} + 4 \times \frac{1}{2} {[x \sqrt{2 - x^{2}} + 2 \sin^{- 1} (\frac{x}{\sqrt{2}})]}_{1}^{\sqrt{2}}$

$= \frac{8}{3} + 2 [2 \times \frac{π}{2} - 1 - 2 \times \frac{π}{4}]$

$= \frac{8}{3} + 2 π - 2 - π$

$= π + \frac{2}{3} sq. units$

Want Us to Email you About Special Offers & Updates?