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

Question

Quizplus · Accepted Answer

We use the formula for the surface area of revolution about the $x$-axis:
$$A = 2\pi\int_a^b f(x)\sqrt{1+\left(\frac{dy}{dx}\right)^2}dx$$
The curve given is $y=2/x$ on $[1,5]$. To use the formula, we first need to find $\frac{dy}{dx}$. Taking the derivative with respect to $x$, we get:
$$\frac{dy}{dx}=-\frac{2}{x^2}$$
Now we can substitute into the formula and simplify:
$$A=2\pi\int_1^5\frac{2}{x}\sqrt{1+\left(-\frac{2}{x^2}\right)^2}dx$$
$$=4\pi\int_1^5\frac{\sqrt{x^4+4}}{x^3}dx$$
Therefore, the answer is choice C.

Write an Integral Giving the Area of the Surface Obtained xxx

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