Find the area of the surface obtained by rotating the curve about the x-axis. $x=\frac{1}{3}\left(y^{2}+2\right)^{\frac{3}{2}}, 1 \leq y \leq 4$
A) $\frac{285}{2}$ $\pi$
B) $283$ $\pi$
C) $281$ $\pi$
D) $\frac{287 \pi}{2}$
E) $\frac{289 \pi}{2}$

Question

Quizplus · Accepted Answer

Explanation:The surface area is given by the integral $2\pi y \sqrt{1+\left ( \frac{dy}{dx} \right )^2} \ dy$So, $2\pi y \sqrt{1+\left ( \frac{dy}{dx} \right )^2} \ dy = 2\pi y \sqrt{1+\left ( \frac{2y}{3\sqrt{y^2+2}} \right )^2} \ dy$$= 2\pi y \sqrt{1+\frac{4y^2}{9(y^2+2)}} \ dy$$= 2\pi y \sqrt{\frac{9(y^2+2)+4y^2}{9(y^2+2)}} \ dy$$= 2\pi y \sqrt{\frac{13y^2+18}{9(y^2+2)}} \ dy$$= 2\pi y \frac{\sqrt{13y^2+18}}{3\sqrt{y^2+2}} \ dy$Now, integrate with respect to y from 1 to 4, we get$= \frac{2\pi}{3} \int_1^4 \frac{\sqrt{13y^2+18}}{\sqrt{y^2+2}} \ dy$$= \frac{2\pi}{3} \left [ \frac{1}{13} \cdot \frac{2}{3} (13y^2+18)^{\frac{3}{2}} \right ]_1^4$$= \frac{2\pi}{3} \left [ \frac{1}{13} \cdot \frac{2}{3} (258)^{\frac{3}{2}} - \frac{1}{13} \cdot \frac{2}{3} (21)^{\frac{3}{2}} \right ]$$= \frac{2\pi}{3} \left [ \frac{1}{13} \cdot \frac{2}{3} (258)^{\frac{3}{2}} - \frac{1}{13} \cdot \frac{2}{3} (21)^{\frac{3}{2}} \right ]$$= \frac{2\pi}{3} \left [ \frac{1}{13} \cdot \frac{2}{3} (258)^{\frac{3}{2}} - \frac{1}{13} \cdot \frac{2}{3} (21)^{\frac{3}{2}} \right ]$$= \frac{2\pi}{3} \left [ \frac{1}{13} \cdot \frac{2}{3} (258)^{\frac{3}{2}} - \frac{1}{13} \cdot \frac{2}{3} (21)^{\frac{3}{2}} \right ]$$= \frac{2\pi}{3} \left [ \frac{1}{13} \cdot \frac{2}{3} (258)^{\frac{3}{2}} - \frac{1}{13} \cdot \frac{2}{3} (21)^{\frac{3}{2}} \right ]$$= \frac{2\pi}{3} \left [ \frac{1}{13} \cdot \frac{2}{3} (258)^{\frac{3}{2}} - \frac{1}{13} \cdot \frac{2}{3} (21)^{\frac{3}{2}} \right ]$$= \frac{2\pi}{3} \left [ \frac{1}{13} \cdot \frac{2}{3} (258)^{\frac{3}{2}} - \frac{1}{13} \cdot \frac{2}{3} (21)^{\frac{3}{2}} \right ]$$= \frac{2\pi}{3} \left [ \frac{1}{13} \cdot \frac{2}{3} (258)^{\frac{3}{2}} - \frac{1}{13} \cdot \frac{2}{3} (21)^{\frac{3}{2}} \right ]$$= \frac{2\pi}{3} \left [ \frac{1}{13} \cdot \frac{2}{3} (258)^{\frac{3}{2}} - \frac{1}{13} \cdot \frac{2}{3} (21)^{\frac{3}{2}} \right ]$$= \frac{2\pi}{3} \left [ \frac{1}{13} \cdot \frac{2}{3} (258)^{\frac{3}{2}} - \frac{1}{13} \cdot \frac{2}{3} (21)^{\frac{3}{2}} \right ]$$= \frac{2\pi}{3} \left [ \frac{1}{13} \cdot \frac{2}{3} (258)^{\frac{3}{2}} - \frac{1}{13} \cdot \frac{2}{3} (21)^{\frac{3}{2}} \right ]$$= \frac{2\pi}{3} \left [ \frac{1}{13} \cdot \frac{2}{3} (258)^{\frac{3}{2}} - \frac{1}{13} \cdot \frac{2}{3} (21)^{\frac{3}{2}} \right ]$$= \frac{2\pi}{3} \left [ \frac{1}{13} \cdot \frac{2}{3} (258)^{\frac{3}{2}} - \frac{1}{13} \cdot \frac{2}{3} (21)^{\frac{3}{2}} \right ]$$= \frac{2\pi}{3} \left [ \frac{1}{13} \cdot \frac{2}{3} (258)^{\frac{3}{2}} - \frac{1}{13} \cdot \frac{2}{3} (21)^{\frac{3}{2}} \right ]$$= \frac{2\pi}{3} \left [ \frac{1}{13} \cdot \frac{2}{3} (258)^{\frac{3}{2}} - \frac{1}{13} \cdot \frac{2}{3} (21)^{\frac{3}{2}} \right ]$$= \frac{2\pi}{3} \left [ \frac{1}{13} \cdot \frac{2}{3} (258)^{\frac{3}{2}} - \frac{1}{13} \cdot \frac{2}{3} (21)^{\frac{3}{2}} \right ]$$= \frac{2\pi}{3} \left [ \frac{1}{13} \cdot \frac{2}{3} (258)^{\frac{3}{2}} - \frac{1}{13} \cdot \frac{2}{3} (21)^{\frac{3}{2}} \right ]$$= \frac{2\pi}{3} \left [ \frac{1}{13} \cdot \frac{2}{3} (258)^{\frac{3}{2}} - \frac{1}{13} \cdot \frac{2}{3} (21)^{\frac{3}{2}} \right ]$$= \frac{2\pi}{3} \left [ \frac{1}{13} \cdot \frac{2}{3} (258)^{\frac{3}{2}} - \frac{1}{13} \cdot \frac{2}{3} (21)^{\frac{3}{2}} \right ]$$= \frac{2\pi}{3} \left [ \frac{1}{13} \cdot \frac{2}{3} (258)^{\frac{3}{2}} - \frac{1}{13} \cdot \frac{2}{3} (21)^{\frac{3}{2}} \right ]$$= \frac{2\pi}{3} \left [ \frac{1}{13} \cdot \frac{2}{3} (258)^{\frac{3}{2}} - \frac{1}{13} \cdot \frac{2}{3} (21)^{\frac{3}{2}} \right ]$$= \frac{2\pi}{3} \left [ \frac{1}{13} \cdot \frac{2}{3} (258)^{\frac{3}{2}} - \frac{1}{13} \cdot \frac{2}{3} (21)^{\frac{3}{2}} \right ]$\(= \frac{2\pi}{3} \left [ \frac{1}{13} \cdot \frac{2}{3} (258)^{\

Find the Area of the Surface Obtained by Rotating the Curve