Chapter 8.3, Problem 55E

To determine

To find : the area of the surface.

Answer to Problem 55E

The area of the surface is $13.614$ .

Explanation of Solution

Given information : The curve about the y-axis $x=\sqrt{y},0\le y\le 2$ .

Calculation :

The equation for finding the surface area or an equation rotated around the y-axis.

  $\begin{array}{l}2\pi {\displaystyle \underset{c}{\overset{d}{\int }}f\left(y\right)\sqrt{1+f\text{'}{\left(y\right)}^2}\times dy}\\ 2\pi {\displaystyle \underset{0}{\overset{2}{\int }}\sqrt{y}\sqrt{1+f\text{'}{\left(y^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\right)}^2}\times dy}\\ 2\pi {\displaystyle \underset{0}{\overset{2}{\int }}\sqrt{y}\sqrt{1+{\left(5y^{-5}\right)}^2}\times dy}\\ 2\pi {\displaystyle \underset{0}{\overset{2}{\int }}\sqrt{y}\sqrt{1+\left(25y^1\right)}\times dy}\\ \approx 13.614\end{array}$