Question

The area of the loop of the curve x²− (y − 1)y² = 0 is equal to

• A. $\frac { 8 } { 15 }$ sq. units
• B. $\frac { 2 } { 15 }$ sq. units
• C. $\frac { 4 } { 15 }$sq. units
• D. None of these

Answer 1

Answer: (A)

$\frac { 8 } { 15 }$ sq. units

Answer 2

We have, x² = (1 − y)y² ⇒ 1 − y ≥ 0

⇒ y ≤ 1. Thus, for the loop, y ∈ [0, 1]. Area of required loop = $2 \int _ { 0 } ^ { 1 } y \sqrt { 1 - y } d y$

$= 2 \int _ { 0 } ^ { 1 } ( 1 - y ) \sqrt { y } d y$

$= 2 \int _ { 0 } ^ { 1 } \left( \sqrt { y } - y ^ { 3 / 2 } \right) d y = 2 \left( \frac { 2 } { 3 } y ^ { 3 / 2 } - \frac { 2 } { 5 } y ^ { 5 / 2 } \right) _ { 0 } ^ { 1 }$

$= \frac { 8 } { 15 }$ sq. units.