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Unit-IV, , Length of Plane Curves: Introduction-Expression for the length of curves, , y = f (x)-Expression for the length of arcs x = f(y);x =f(Oiy = WO);, , r=f(8), , Volumes and Surface of Revolution: Introduction-Expression for the volume obtained, by revolving about either axis - Expression for the volume obtained by revolving about, any line —Area of the surface of the frustum of a cone —Expression for the surface of, revolution —-Pappus Theorem-Surface of revolution

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Rectification: The process of determining the length of the arc, , of the plane curves is defined as rectification, , Expression for the length of the curve y = f(x):, Theorem: The length of the arc of the curve y = f(x), Included between two points whose abscissae, Z, are a and bis aC 1+ (2) dx, Proof: Let A and B be two points with abscissae a, b on the, Curve y = f(x), if s denote the length of the arc of the curve included, between a fixed point A and a variable point P whose, , abscissa of x so that it is a function of x, ds _ dy 2 _, we have ae ll st (2) (1), Integrating on both sides of eq.(1) w.r.t. x from a to b, , i 1+ (2) dx = Pas ay, , ax ~ Ja ax

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= [sla, , = s(b) — s(a), = (The value of S for x = b) — (The value of S for x = a), , = Arc AB -—0, , i. 1+ (2) ax = Are AB, , 2, Hence length of arc of the curve = i 1+ (2) dx

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Length of plane curves: The expression for the length of the, , arc for different forms of plane curves is given below, (i).Cartesian Form:, , (a) The length of the arc of the curve y= f(x) included, , between two points whose abscissa are x =a andx =b, , is s= ey 1+ (2 dx, , (b) The length of the arc of the curve x = f(y) included, , between two points whose ordinates are y=candy=d, , is s= pre 1+ ey dy

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(ii).Parametric Form:, The length of the arc of the curve x= f(@),y= (6), , included between two points whose parametric values are, , 86=aandd0=f is, s= Ice \((G) +(@) |e, , (iii). Polar Form:, (a) The length of the arc of the curve r = f(@) included, , between two points whose vertical angles are 0=a, , and @ =8 is s= fpf |r? +(2) *| a0, , (6) The length of the arc of the curve @ = f(r) included, , between two points whose radii are r=aandr=b, , is nn [1+ (2 late