Thomas’ Calculus Early Transcendentals 13ed

Volume and Surface Area of Revolution

Notation:

a: axial variable. r: radial variable. The generating region:

R = {(a, r), α < a < β, 0 < r_{1}(a) < r < r_{2}(a)}

= {(a, r), γ < r < δ, 0 < a_{1}(r) < a < a_{2}(r)}

Volume of revolution:

Method of disks:

V = ∫_{α}^{β} π(r_{2}^{2}(a) − r_{1}^{2}(a))da

Method of cylindrical shells:

V = ∫_{γ}^{δ} 2πr(a_{2}(r) − a_{1}(r))da

Arclength:

ds = \sqrt{dx^{2} + dy^{2}},

L = ∫_{α}^{β} \sqrt{1 + (\frac{dy}{dx})^{2}}dx = ∫_{α}^{β} \sqrt{1 + (f′(x))^{2}}dx, if y = f(x)

or

L = ∫_{γ}^{δ} \sqrt{1 + (\frac{dx}{dy})^{2}}dy = ∫_{γ}^{δ} \sqrt{1 + (g′(y))^{2}}dy, if x = g(y)

Surface of revolution:

a: axial variable. r: radial variable.

case 1:

The generating curve is r = f(a) on α < a < β:

A = ∫_{α}^{β} 2πrds = ∫_{α}^{β} 2πf(a)\sqrt{1 + (f′(a))^{2}}da

case 2:

The generating curve is a = g(r) on γ < r < δ:

A = ∫_{γ}^{δ} 2πrds = ∫_{γ}^{δ} 2πr\sqrt{1 + (g′(r))^{2}}dr