The area (in sq. units) of the region {(x, y): y2 ≥ 2x and x2 + y2 ≤ 4x, x ≥ 0, y ≥ 0} is

Engineering

Mathematics

Introduction to Area

Question

The area (in sq. units) of the region {(x, y): y² ≥ 2x and x² + y² ≤ 4x, x ≥ 0, y ≥ 0} is

$\frac{π}{2} - \frac{2 \sqrt{2}}{3}$

$π - \frac{4}{3}$

$π - \frac{8}{3}$

$π - \frac{4 \sqrt{2}}{3}$

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x² + y² $\leq$ 4x ⇒ (x – 2)² + y² $\leq$ 4

y² $\geq$ 2x

Point of intersection = x² + 2x = 4x

⇒ x² – 2x = 0

⇒ x = 0 or x = 2

Area bounded

$= {(\frac{1}{4})}^{th} area of circle - \int_{0}^{2} \sqrt{2 x} dx$

$= \frac{1}{4} \times π \cdot 2^{2} - {(\frac{\sqrt{2} x^{\frac{3}{2}}}{\frac{3}{2}})}_{0}^{2} = π - \frac{8}{3} .$

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