14.4. Lengths of curves and surfaces of revolution. Introduction. Prerequisites. Learning Outcomes

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1 Lengths of curves nd surfces of revolution 4.4 Introduction Integrtion cn be used to find the length of curve nd the re of the surfce generted when curve is rotted round n xis. In this section we stte nd use the formule for doing this. Prerequisites Before strting this Section you should... Lerning Outcomes After completing this Section you should be ble to... be ble to clculte definite integrls find the length of number of curves find the re of the surfce generted when curve is rotted bout n xis

2 . The length of curve To find the length of curve in the xy plne we first divide the curve into lrge number of pieces. We mesure (or, t lest, pproximte) the length of ech piece nd then by n obvious summtion process obtin n estimte for the length of the curve. Theoreticlly, we llow the number of pieces to increse without bound, implying tht the length of ech piece will tend to zero. In this limit the summtion process becomes n integrtion process. y y(x) δy b x Figure Figure shows the portion of the curve y(x) between x = nd x = b. A smll piece of this curve hs been selected nd cn be considered s the hypotenuse of tringle with bse nd height δy. (Here nd consequently, δy re intended to be smll so tht the curved piece cn be regrded s stright piece). Using Pythgors theorem, the length of the hypotenuse is: + δy = x=b lim x= + ( ) δy By summing ll such contributions between x = nd x = b, nd letting weobtin n expression for the totl length of the curve: ( ) δy + But we lredy know how to write such n expression in terms of n integrl. We obtin the following result: Key Point Given curve with eqution y = f(x), then the length of the curve between the points where x = nd x = b is given by the formul: b ( ) dy + HELM (VERSION : Mrch 8, 4): Workbook Level 4.4: Lengths of curves nd surfces of revolution

3 Becuse of the form of the integrnd, nd in prticulr the squre root, integrls of this type re often difficult to clculte nd in prctice, pproximte rther thn exct methods re normlly needed to perform the integrtion. We shll illustrte the ppliction of the formul by n exmple which could be clculted in much simpler wy, before looking t some hrder exmples. Find the length of the curve y =3x +between x =nd x =5. In this exmple, the curve is in fct stright line segment, nd its length could be obtined using methods other thn integrtion. Notice from the formul in the Key Point tht it is necessry to find dy.dothis first. dy =3 Applying the formul we find length of curve = 5 5 +(3) = [ ] 5 = x = (5 ) =4 =.65 Thus the length of the curve y =3x +between the points where x =nd x =5is.65 units. Find the length of the curve y = cosh x between x =nd x =shown in Figure. y y = cosh x x Figure 3 HELM (VERSION : Mrch 8, 4): Workbook Level 4.4: Lengths of curves nd surfces of revolution

4 First write down dy dy = dy = sinh x Hence write down the required integrl: +sinh x This integrl cn be evluted by mking use of the hyperbolic identity cosh x sinh x = Write down the integrl which results fter pplying this identity. cosh x Perform the integrtion for yourself to find the required length. [sinh x] =3.63 Thus the length of y = cosh x between x =nd x =is 3.63 units. The finl exmple is more complicted still nd requires the use of hyperbolic substitution nd knowledge of the hyperbolic identities. Find the length of the curve y = x between x =nd x =3. Given y = x then dy =x. Apply the formul to obtin the integrl required: 3 +4x HELM (VERSION : Mrch 8, 4): Workbook Level 4.4: Lengths of curves nd surfces of revolution 4

5 Mke the substitution x = sinh u, = cosh u, toobtin n integrl in terms of u. du sinh 6 +sinh u cosh u du Use the hyperbolic identity cosh u sinh u =to rewrite this integrl: sinh 6 cosh u du Another hyperbolic identity is cosh u = (cosh u + ). integrnd. Apply this identity to rewrite the 4 sinh 6 (cosh u +)du Finlly, performing the integrtion we cn now complete the clcultion: sinh 6 (cosh u +)du = [ ] sinh sinh u 6 + u 4 4 = 9.75 Thus the length of the curve y = x between x =nd x =3is 9.75 units. Exercises. Find the length of the line y =x +7between x =nd x =3using the technique of this section. Verify your result from your knowledge of the stright line.. Find the length of y = x 3/ between x =nd x =5. 3. Clculte the length of the curve y =4x 3 between x =nd x =. Answers. 5= (first qudrnt only). 5 HELM (VERSION : Mrch 8, 4): Workbook Level 4.4: Lengths of curves nd surfces of revolution

6 . The re of surfce of revolution. In section we found n expression for the volume of solid of revolution. Here we consider the more complicted problem of formulting n expression for the surfce re of solid of revolution. y y(x) ( x, y) δy b x Figure 3 Figure 3 shows the portion of the curve y(x) between x = nd x = b which is rotted round the x xis through 36. A smll disc, of thickness, ofthe solid of revolution hs been selected. Its rdius is y nd so its circumference hs length πy. (As usul we ssume is smll so tht the curved prt of y(x) representing the hypotenuse of the highlighted tringle cn be regrded s stright). This surfce ribbon, shown shded, hs length πy nd width () +(δy) nd so its re is then, to good pproximtion, πy () +(δy).wenow let toobtin the result: Key Point Given curve with eqution y = f(x), then the surfce re of the solid generted by rotting tht prt of the curve between the points where x = nd x = b round the x xis is given by the formul: b ( ) dy re of surfce = πy + Find the re of the surfce generted when the prt of the curve y = x 3 between x =nd x =4is rotted round the x xis. HELM (VERSION : Mrch 8, 4): Workbook Level 4.4: Lengths of curves nd surfces of revolution 6

7 The re of surfce is given by re = = b 4 πy + ( ) dy πx 3 +(3x ) = 4 πx 3 +9x 4 This integrl cn be found by mking substitution u = +9x 4, du =36x3 so tht x 3 = du When x =,u =nd when x =4,u = 35. Write down the corresponding integrl in terms of u. 36. π 8 udu Perform the integrtion. 35 Apply the limits of integrtion to find the re. ]35 3 [ π 8 π 7 ( (35) 3/ ) Exercises. The line y = x between x =nd x =is rotted round the x xis. () Find the re of the surfce generted. (b) Verify this result by finding the curved surfce re of the corresponding cone. (The curved surfce re of cone of rdius r nd slnt height l is πrl.). Find the re of the surfce generted when y = x for x isrotted completely bout the x xis. Answers. π HELM (VERSION : Mrch 8, 4): Workbook Level 4.4: Lengths of curves nd surfces of revolution