Volumes of solids of revolution


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1 Volumes of solids of revolution We sometimes need to clculte the volume of solid which cn be obtined by rotting curve bout the xxis. There is strightforwrd technique which enbles this to be done, using integrtion. In order to mster the techniques explined here it is vitl tht you undertke plenty of prctice exercises so tht they become second nture. After reding this text, nd/or viewing the video tutoril on this topic, you should be ble to: find the volume of solid of revolution obtined from simple function y f(x) between given limits x nd x b; find the volume of solid of revolution obtined from simple function y f(x) where the limits re obtined from the geometry of the solid. Contents 1. Introduction 2 2. The volume of sphere 4. The volume of cone 4 4. Another exmple 5 5. Rotting curve bout the yxis 6 1 c mthcentre April 27, 2008
2 1. Introduction Suppose we hve curve, y f(x). y f(x) x x b Imgine tht the prt of the curve between the ordintes x nd x b is rotted bout the xxis through 60. The curve would then mp out the surfce of solid s it rotted. Such solids re clled solids of revolution. Thus if the curve ws circle, we would obtin the surfce of sphere. If the curve ws stright line through the origin, we would obtin the surfce of cone. Now we lredy know wht the formule for the volumes of sphere nd cone re, but where did they come from? How cn they clculted? If we could find generl method for clculting the volumes of the solids of revolution then we would be ble to clculte, for exmple, the volume of sphere nd the volume of cone, s well s the volumes of more complex solids. To see how to crry out these clcultions we look first t the curve, together with the solid it mps out when rotted through 60. y f(x) Now if we tke crosssection of the solid, prllel to the yxis, this crosssection will be circle. But rther thn tke crosssection, let us tke thin disc of thickness δx, with the fce of the disc nerest the yxis t distnce x from the origin. c mthcentre April 27,
3 y f(x) x y y + δy δx x x b The rdius of this circulr fce will then be y. The rdius of the other circulr fce will be y + δy, where δy is the chnge in y cused by the smll positive increse in x, δx. The disc is not cylinder, but it is very close to one. It will become even closer to one s δx, nd hence δy, tends to zero. Thus we pproximte the disc with cylinder of thickness, or height, δx, nd rdius y. The volume δv of the disc is then given by the volume of cylinder, πr 2 h, so tht δv y 2 δx. So the volume V of the solid of revolution is given by xb lim δv δx 0 x xb lim πy 2 δx δx 0 x πy 2 dx, where we hve chnged the limit of sum into definite integrl, using our definition of integrtion. This formul now gives us wy to clculte the volumes of solids of revolution bout the xxis. Key Point If y is given s function of x, the volume of the solid obtined by rotting the portion of the curve between x nd x b bout the xxis is given by πy 2 dx. c mthcentre April 27, 2008
4 2. The volume of sphere The eqution x 2 + y 2 r 2 represents the eqution of circle centred on the origin nd with rdius r. So the grph of the function y r 2 x 2 is semicircle. r r y r 2 x 2 We rotte this curve between x r nd x r bout the xxis through 60 to form sphere. Now x 2 + y 2 r 2, nd so y 2 r 2 x 2. Therefore r πy 2 dx (r 2 x 2 ) dx r ] r [r 2 x x r ) {(r r 4πr This is the stndrd result for the volume of sphere.. )} ( r + r. The volume of cone Suppose we hve cone of bse rdius r nd verticl height h. We cn imgine the cone being formed by rotting stright line through the origin by n ngle of 60 bout the xxis. θ h r c mthcentre April 27,
5 The grdient of the stright line is tn θ, nd from the rightngled tringle we see tht tn θ r/h. Thus the eqution of the line is y rx/h, nd the limits of integrtion re from x 0 to x h. So h 0 h 0 πy 2 dx ( rx ) 2 π dx h π r2 x 2 dx h 2 ] h [ r 2 x h 2 0 ( ) r 2 h h 0 2 r2 h This is the stndrd result for the volume of cone.. 4. Another exmple The curve y x 2 1 is rotted bout the xxis through 60. Find the volume of the solid generted when the re contined between the curve nd the xxis is rotted bout the xxis by 60. From the wording of the question, portion of the curve trps n re between itself nd the xxis. Hence the curve must cross the xxis. To find the points where this hppens we need to set y 0. So we need to solve the eqution x Fctorising, (x 1)(x + 1) 0, nd therefore x 1 or x. Here is sketch of the curve. 1 The grph of y x c mthcentre April 27, 2008
6 We clculte the volume s follows. 1 1 πy 2 dx π(x 2 1) 2 dx [ x 5 5 2x 16π 15. (x 4 2x 2 + 1) dx ] 1 + x ) {( ( )} 1 5. Rotting curve bout the y xis We hve looked t how to find the volume of solid creted by rotting n re bout the xxis. But we cn lso rotte n re bout the yxis. How cn we find the volume in this cse? x f(y) y d y c To crry out such clcultion, we must interchnge the rôles of x nd y. First, the eqution of the curve must be given s x f(y), rther thn s y f(x). And secondly, the limits must be given in terms of y, s y c nd y d. Thus the formul for the volume becomes d c πx 2 dy. c mthcentre April 27,
7 Exercises 1. Find the volume of the solid of revolution generted when the re described is rotted bout the xxis. () The re between the curve y x nd the ordintes x 0 nd x 4. (b) The re between the curve y x /2 nd the ordintes x 1 nd x. (c) The re between the curve x 2 + y 2 16 nd the ordintes x nd x 1. (d) The re between the curve x 2 y 2 9 nd the ordintes x 4 nd x. (e) The re between the curve y (2 + x) 2 nd the ordintes x 0 nd x The re between the curve y 1/x, the yxis nd the lines y 1 nd y 2 is rotted bout the yxis. Find the volume of the solid of revolution formed.. The re between the curve y x 2, the yxis nd the lines y 0 nd y 2 is rotted bout the yxis. Find the volume of the solid of revolution formed. 4. The re cut off by the xxis nd the curve y x 2 x is rotted bout the xxis. Find the volume of the solid of revolution formed. 5. Sketch the curve y 2 x(x 4) 2 nd find the volume of the solid of revolution formed when the closed loop of the curve is rotted bout the xxis. 6. A conicl funnel is formed by rotting the curve y 1 x bout the yxis. The rdius of the rim of the funnel is to 6 cm. Find the depth of the funnel nd its volume. Answers 1. () 21 1 π (b) 20π (c) 11 π (d) 1 π (e) π π. 2π π π 6. 24π 7 c mthcentre April 27, 2008
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