Question

Area of loop of curve y² = x(1 – x)² is :

• A. $\frac{8}{15}$sq. units
• B. $\frac{7}{15}$ sq. units
• C. $\frac{4}{15}$ sq. units
• D. None

Answer 1

Answer: (A)

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

Answer 2

Curve is symmetric about x – axis

Reqd. Area = 2 $\int_{0}^{1}\sqrt{x}$ (1 – x) dx = 2$\int_{0}^{1}\sqrt{x}$ – x^3/2 dx

= 2 $\left[ \frac { 2 \mathrm { x } ^ { 3 / 2 } } { 3 } - \frac { 2 \mathrm { x } ^ { 5 / 2 } } { 5 } \right] _ { 0 } ^ { 1 }$

= 2$\left( \frac{2}{3} - \frac{2}{5} \right)$ = 2$\left( \frac{10 - 6}{15} \right)$ = 8/15