Chapter 8.3, Problem 57E

To determine

To find : the area of the surface.

Answer to Problem 57E

The area of the surface is $16.110$ .

Explanation of Solution

Given information : The curve about the y-axis $x=y^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}-{\left(\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.},1\le y\le 3$ .

Calculation :

Let $g\left(y\right)=y^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}-{\left(\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$

Then, the surface area is given by:

  $\begin{array}{l}S=2\pi {\displaystyle \underset{c}{\overset{d}{\int }}f\left(y\right)\sqrt{1+f\text{'}{\left(y\right)}^2}\times dy}\\ S=2\pi {\displaystyle \underset{1}{\overset{3}{\int }}{y}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}-{\left(\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{\sqrt{1+\left(\frac{1}{2\sqrt{y}}\right)}}^{2}dy}\\ S\approx 16.110\end{array}$