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Unit Four B.Tech Ist Semester Content Arc length Surface area of solid of revolution. Volume of solid of revolution. Arc Length Length of arc s of the Parametric curve y=f(x) between the points x=a and x=b is b dy 2 s=∫ 1 + dx a dx Polar curve r=f(θ) between the points θ=α and θ=β β 2 dr 2 s=∫ r + dθ α dθ Parametric curve x=f(t), y=f(t) between the ponts t=a and t=b is b dx dy 2 2 s=∫ + dt a dt dt e.g.: (i) Find the length of arc of the parabola x2=4ay measured from the vertex to one extremity of the latus –rectum. [ ( Ans. a 2 + log 1 + 2 )] (ii) Find the entire length of the cardioid r=a(1+cosθ). Ans . 2a Surface area of solid of revolution (i) About the x-axis: The surface area of the solid generated by the revolution about x-axis, of the arc of the curve y=f(x) from x=a to x=b. b ∫ 2πy ds a (ii) About the y-axis: The surface area of the solid generated by the revolution about x-axis, of the arc of the curve x=f(y) from y=a to y=b. b ∫ 2πx ds a For polar-coordinates (iii) About the initial line: The surface area of the solid generated by the revolution about initial line, of the arc of the curve r=f(θ) bonded by θ=α and θ=β is β ∫ 2π r sin θ ds α (iv) About the line θ=π/2: The surface area of the solid generated by the revolution about line θ=π/2, of the arc of the curve r=f(θ) bonded by θ=α and θ=β is β ∫ 2π r cosθ ds α e.g. : (i) Find the surface area of the solid generated by the revolution of the asteroid x=a cos3t , y=a sin3t, about the y-axis. 12 Ans . π a2 5 (ii) Find the surface of the solid formed by revolving the cardioid r=a(l + cosθ) about the initial line. 32π a 2 Ans . 5 Volume of solid of revolution About the x-axis: The volume of the solid generated by the revolution about the x-axis, of the area bounded by the curve y=f(x), the x-axis and the ordinates x=a, x=b is b ∫ π y dx 2 a About the y-axis: The volume of the solid generated by the revolution about the y-axis, of the area bounded by the curve x=f(y), the y-axis and the ordinates y=a, y=b is b ∫ π x dy 2 a Volume of solid of revolution About any axis: The volume of the solid generated by the revolution about the any axis LM of area bounded by the curve AB, the axis LM and the perpendicular AL,BM on the axis, is OM B ∫ OL π ( PN ) 2 d (ON ) A P O L N M e.g.: (i) Find the volume of the solid formed by the revolution of the cissoid y3(2a-x)=x3 about its asymptote. Ans . 2π 2 a 3 (ii) Find the volume generated by the revolution of the area bounded by x-axis, the centenary y=c cosh x/c and the ordinates x=±c, about axis of x. Ans . π c 3 (1 + sinh 1 cosh 1) Volume of solid of revolution for polar curve: The volume of the solid generated by the revolution of the area bounded by the curve r=f(θ) and the redii vector θ=α, θ=β (a) about initial line θ=0 is β 2 3 ∫ 3 π r sin θ dθ α (b) about initial line θ=π/2 is β 2 3 ∫ 3 π r cosθ dθ α Volume Between two curve The volume of the solid generated by the revolution about the x-axis of the area bounded by the curves y=f(x), y=g(x), and the ordinates x=a, x=b is b ∫π (y 2 2 1 − y2 ) dx a where y1 is the ‘y’ of the upper curve and y2 that of the lower curve. e.g.: (i) Find the volume of the solid formed by revolving a loop of lemniscate r2=a2 cos 2θ about the initial line. π a3 1 1 Ans . 4 2 ( log 2 + 1 − 3 ) (ii) The area bounded by the parabola y2=4x and the straight line 4x-3y+2=0 is rotated about the y-axis, obtained the volume of the solid formed by this revolution. π Ans . 20

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posted: | 10/27/2012 |

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