Answer
$\dfrac{471,295 \pi}{1024}$
Work Step by Step
Given: $x=4 \sqrt t$ $\implies$ $\frac{dx}{dt}=\frac{2}{ \sqrt t}$
$y=\frac{t^3}{3}+\frac{1}{2t^2}$ $\implies$ $y=t^2-\frac{1}{t^3}$
To calculate the area of the surface, we will use the equation:
$A=2 \pi \int_1^4 y(t) \sqrt{(dx/dt)^2+(dy/dt)^2}dt$
$=2 \pi \int_1^4 (\frac{t^3}{3}+\frac{1}{2t^2}) \sqrt{(\frac{2}{ \sqrt t})^2+(t^2-\frac{1}{t^3})^2}dt$
$=2 \pi \int_1^4 [\frac{t^3}{3}+\frac{1}{2} +\frac{1}{3}+\frac{1}{2t^5}]dt$
$=2 \pi \int_1^4 [\frac{t^5}{3}+\frac{5}{6} +\frac{1}{2t^5}]dt$
$=2 \pi[\frac{t^6}{18}+\frac{5t}{6} -\frac{1}{8t^4}]_1^4$
$=\dfrac{471,295 \pi}{1024}$
