Approximate the area of the surface generated by revolving the curve y = sin x, , about the x-axis. Approximate area to three decimal places.
A) 14.424
B) 8.439
C) 7.212
D) 6.912
E) 3.606

Question

Approximate the area of the surface generated by revolving the curve y = sin x,  , about the x-axis. Approximate area to three decimal places.
A) 14.424 
B) 8.439 
C) 7.212 
D) 6.912 
E) 3.606

Quizplus · Accepted Answer

The formula for the surface area of revolution about the x-axis is:
$$S = 2\pi\int_{a}^{b}y\sqrt{1+\left(\frac{dy}{dx}\right)^2}dx$$
We have $y=\sin x$ and $\frac{dy}{dx} = \cos x$. Thus, we need to evaluate the integral:
$$S = 2\pi\int_{0}^{2\pi}\sin x\sqrt{1+\cos^2 x}dx$$
This is a challenging integral to evaluate exactly, but we can use numerical methods or a graphing calculator to get an approximate answer. Using a graphing calculator, we get:
$$S \approx 7.212$$
Therefore, the best choice is C.

Approximate the Area of the Surface Generated by Revolving the Curve