(a) In the discussion associated with Exercises 7580 ofSection 10.1, formulas were given

Chapter 10, Problem 64

(choose chapter or problem)

(a) In the discussion associated with Exercises  of Section , formulas were given for the area of the surface of revolution that is generated by revolving a parametric curve about the -axis or -axis. Use those formulas to derive the following formulas for the areas of the surfaces of revolution that are generated by revolving the portion of the polar curve \(r=f(\theta)\) from \(\theta=\alpha\) to \(\theta=\beta\) about the polar axis and about the line \(\theta=\pi / 2\) \(S=\int_{\alpha}^{\beta} 2 \pi r \sin \theta \sqrt{r^{2}+\left(\frac{d r}{d \theta}\right)^{2}} d \theta\) About \(\theta=0\) \(\int_{a}^{\beta} 2 \pi r \cos \theta \sqrt{r^{2}+\left(\frac{d r}{d \theta}\right)^{2}} d \theta\) About \(\theta=\pi / 2\)
(b) State conditions under which these formulas hold.

Equation Transcription:

Text Transcription:

r=f(theta)

theta = alpha

theta = beta

theta = pi/2

S=integral _alpha ^beta 2pi r sin theta sqrt r^2 +(dr/d theta)^2 d theta

theta = 0

integral _alpha ^beta 2pi r cos theta sqrt r^2 + (dr/d theta)^2 d theta