Question

The area of the loop of the curve $y^2=x^4(x+2)$ is [in square units]

• A. $\frac{32\sqrt{2}}{105}$
• B. $\frac{64\sqrt{2}}{105}$
• C. $\frac{128\sqrt{2}}{105}$
• D. $\frac{256\sqrt{2}}{105}$

Answer 1

The correct option is D

$\frac{256\sqrt{2}}{105}$

Area $=2{\int }_{-2}^{0}ydx=2{\int }_{-2}^0{x}^2\sqrt{x+2}dx$

$=4\sqrt{2}{\int }_0^{\sqrt{2}}(z^2-2{)}^2{z}^{2}dz(where\sqrt{x+2}=z)$

$=4{[\frac{z^7}{7}-\frac{4z^5}{5}+\frac{4z^3}{3}]}_0^{\sqrt{2}}$

$=\frac{256\sqrt{2}}{105}$