Write an integral giving the area of the surface obtained by revolving the curve about the $x$-axis. (Do not evaluate the integral.)
$$y = \frac { 2 } { x } \text { on } [ 4,5 ]$$
A) $2 \pi \int _ { 4 } ^ { 5 } \frac { 2 } { x } \left( 1 + \left( - \frac { 2 } { x ^ { 2 } } \right) ^ { 2 } \right) d x$
B) $2 \pi \int _ { 4 } ^ { 5 } \frac { \sqrt { x ^ { 4 } - 4 } } { x ^ { 3 } } d x$
C) $4 \pi \int _ { 4 } ^ { 5 } x ^ { 3 } \sqrt { x ^ { 4 } + 4 } d x$
D)
$4 \pi \int _ { 4 } ^ { 5 } \frac { \sqrt { x ^ { 4 } + 4 } } { x ^ { 3 } } d x$

Question

Quizplus · Accepted Answer

The formula to find the surface area of a curve revolved about the $x$-axis is given by $$S = 2 \pi \int_a^b y\sqrt{1+\left(\frac{dy}{dx}\right)^2}dx.$$ Using this formula and substituting the given values, we get the integral as $$S = 2 \pi \int_4^5 \frac{2}{x}\sqrt{1+\left(\frac{2}{x^2}\right)^2}dx = 4 \pi \int_4^5 \frac{\sqrt{x^4+4}}{x^3}dx,$$ which is option D.

Write an Integral Giving the Area of the Surface Obtained xxx

Write an Integral Giving the Area of the Surface Obtained $x$