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Let C be the curve x = f(t), y = g(t), for ast≤ b, where f' and g' are continuous on [a, b] and C does not intersect itself, except possible at its endpoints. If g is nonnegative on [a, b], the area of the surface obtained by revolving C about the x-axis is S = 2r g(t)√f(t)² + g'(t)² dt. Likewise, if f is nonnegative a on [a, b], then the area of the surface obtained by revolving C about the y-axis is S = 2x f(t),√/f' (t)² + g' (t)² dt. a Find the area of the surface obtained by revolving one arch of the cycloid x= 12t - 12 sint, y = 12-12 cost, for 0 st≤ 2, about the x-axis.

Let C be the curve x = f(t), y = g(t), for ast≤ b, where f' and g' are continuous on [a, b] and C does not intersect itself, except possible at its endpoints. If g is nonnegative on [a, b], the area of the surface obtained by revolving C about the x-axis is S = 2r g(t)√f(t)² + g'(t)² dt. Likewise, if f is nonnegative a on [a, b], then the area of the surface obtained by revolving C about the y-axis is S = 2x f(t),√/f' (t)² + g' (t)² dt. a Find the area of the surface obtained by revolving one arch of the cycloid x= 12t - 12 sint, y = 12-12 cost, for 0 st≤ 2, about the x-axis.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Let \( C \) be the curve \( x = f(t) \), \( y = g(t) \), for \( a \leq t \leq b \), where \( f' \) and \( g' \) are continuous on \([a, b]\) and \( C \) does not intersect itself, except possibly at its endpoints. If \( g \) is nonnegative on \([a, b]\), the area of the surface obtained by revolving \( C \) about the x-axis is \[ S = \int_{a}^{b} 2\pi g(t) \sqrt{f'(t)^2 + g'(t)^2} \, dt. \] Likewise, if \( f \) is nonnegative on \([a, b]\), then the area of the surface obtained by revolving \( C \) about the y-axis is \[ S = \int_{a}^{b} 2\pi f(t) \sqrt{f'(t)^2 + g'(t)^2} \, dt. \] Find the area of the surface obtained by revolving one arch of the cycloid \( x = 12t - 12 \sin t \), \( y = 12 - 12 \cos t \), for \( 0 \leq t \leq 2\pi \), about the x-axis.
Transcribed Image Text:Let \( C \) be the curve \( x = f(t) \), \( y = g(t) \), for \( a \leq t \leq b \), where \( f' \) and \( g' \) are continuous on \([a, b]\) and \( C \) does not intersect itself, except possibly at its endpoints. If \( g \) is nonnegative on \([a, b]\), the area of the surface obtained by revolving \( C \) about the x-axis is \[ S = \int_{a}^{b} 2\pi g(t) \sqrt{f'(t)^2 + g'(t)^2} \, dt. \] Likewise, if \( f \) is nonnegative on \([a, b]\), then the area of the surface obtained by revolving \( C \) about the y-axis is \[ S = \int_{a}^{b} 2\pi f(t) \sqrt{f'(t)^2 + g'(t)^2} \, dt. \] Find the area of the surface obtained by revolving one arch of the cycloid \( x = 12t - 12 \sin t \), \( y = 12 - 12 \cos t \), for \( 0 \leq t \leq 2\pi \), about the x-axis.
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Step 1: Introduction of the given problem

x equals 12 t minus 12 sin open parentheses t close parentheses y equals 12 minus 12 cos open parentheses t close parentheses 0 less or equal than t less or equal than 2 pi

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