Mathematical Methods for Physics and Engineering - Matematica.NET

PRELIMINARY CALCULUSf(x)y = f(x)∆s∆x∆yxFigure 2.11 The distance moved along a curve, ∆s, corresponding to thesmall changes ∆x and ∆y.In plane polar coordinates,ds = √ (dr) 2 +(rdφ) 2 ⇒ s =∫ r2r 1√1+r 2 ( dφdr) 2dr.(2.43)◮Find the length of the curve y = x 3/2 from x =0to x =2.Using (2.42) and noting that dy/dx = 3 2√ x, the length s of the curve is given by∫ 2s =[=023= 827√1+ 9 4 xdx( 4)(9 1+9[ ( 112] x) 3/2240) ] 3/2− 1 . ◭[ (1+ ]= 8 9x) 3/2227 40Surfaces of revolutionConsider the surface S formed by rotating the curve y = f(x) about the x-axis(see figure 2.12). The surface area of the ‘collar’ formed by rotating an elementof the curve, ds, about the x-axis is 2πy ds, and hence the total surface area isS =∫ ba2πy ds.Since (ds) 2 =(dx) 2 +(dy) 2 from (2.41), the total surface area between the planesx = a and x = b is√∫ b( ) 2 dyS = 2πy 1+ dx. (2.44)dxa74