The area (in sq. units) of the region A = \left{\right. \left(\right. x , y \left.\right) : \frac{y^{2}}{2} \leq x \leq y + 4 \left.\right} is

Engineering

Mathematics

Introduction to Area

Question

The area (in sq. units) of the region $A = {(x, y) : \frac{y^{2}}{2} \leq x \leq y + 4}$ is

$\frac{53}{3}$

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y² = 2x …..(i)

and x – y – 4 = 0 ………(ii)

solving (1) and (2)

(x – y)² = 2x

⇒ x² – 10x + 16 = 0

⇒ x = 8,2 and y = 4,– 2

$A = \int_{- 2}^{4} (y + 4 - \frac{y^{2}}{2}) dy$

$A = {(\frac{y^{2}}{4} + 4 y - \frac{y^{3}}{6})}_{- 2}^{4}$

$A = (4 + 16 - \frac{64}{6}) - (1 - 8 + \frac{8}{6}) = 18$