Section 9 4 Area of a surface of revolution A surface of revolution is formed when a curve is rotated about a line Let s start with some simple surfaces The lateral surface area of a circular cylinder with base radius r and height h is A 2 rh The lateral surface area of a circular cone with base radius r and slant height l is A rl The lateral surface area of a band frustum of a cone with slant height l upper radius r1 and and lower radius r2 is r1 r2 A 2 rl here r 2 Now we consider the surface which is obtained by rotating the curve y f x a x b about the x axis f x 0 for all x in a b and f 0 x is continuous We take a partition P of a b by points a x0 x1 xn b and let yi f xi so that the point Pi xi yi lies on the curve The part of the surface between xi 1 and xi is approximated by taking the line segment Pi 1 Pi and rotating it about the x axis The result is a band with slant height Pi 1 Pi and average radius r 12 yi 1 yi its surface area is Si 2 yi 1 yi Pi 1 Pi 2 We know that Pi 1 Pi p p xi 2 yi 2 1 f 0 x i 2 xi where x i xi 1 xi Since xi is small we have yi f xi f x i and also yi 1 f xi 1 f x i since f is continuous p Si 2 f x i 1 f 0 x i 2 xi 1 Thus the area of the complete surface is n X SX 2 lim kP k 0 Z 2 b p f x i 1 f 0 x i 2 xi i 1 Z b p 0 2 f x 1 f x dx 2 f x a s dy 1 dx a 2 dx If the curve is described as x g y c y d then the formula for the surface area is s 2 Z d p Z d dx 0 2 SX 2 y 1 g y dy 2 y 1 dy dy c c Let s a curve C is defined by the equations x x t a t b y y t The area of the surface generated by rotating C about x axis is Zb SX 2 s y t dx dt 2 dy dt 2 dt a For rotation about the y axis the surface area formulas are if the curve is given as y f x a x b then the formula for the surface area is Zb SY 2 s dy x 1 dx 2 dx a if the curve is described as x g y c y d then the formula for the surface area is s 2 Z d dx SY 2 g y 1 dy dy c If a curve C is defined by the equations x x t a t b y y t then the area of the surface generated by rotating C about y axis is Zb SY 2 s x t a 2 dx dt 2 dy dt 2 dt Example 1 Find the area of the surface obtained by rotating the curve about x axis a y x 4 x 9 b y 2 4x 4 0 x 8 c x t a cos3 t y t a sin3 t 0 t 2 a is a constant 3 For rotation about the y axis the surface area formulas are if the curve is given as y f x a x b then the formula for the surface area is Zb SY 2 s df x 1 dx 2 dx a if the curve is described as x g y c y d then the formula for the surface area is s 2 Z d dg SY 2 g y 1 dy dy c if the is defined by the equations x x t y y t a t b then the area of the surface is Zb SY s 2 2 dx dy x t dt dt dt a Example 2 Find the area of the surface obtained by rotating the curve about y axis p a x 2y y 2 0 y 1 b y 1 x2 0 x 1 4 c x et t y 4et 2 0 t 1 5
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