Section Volumes by Slicing and Rotation About an Axis
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1 Sectio Volumes y Slicig ad Rotatio Aout a Axis Itroductio Recall that we approximated the area uder a curve y = f HxL y first usig the Riema sum f Ix * k M D x k=1 We otaied the exact value y takig the limit of the Riema sums as Ø. This gave us a defiite itegral. lim Ø f Ix * k M D x = Ÿ f HxL x a k=1 If FHxL was ay atiderivative of f HxL, the y the Fudametal Theorem of Calculus, the shortcut to evaluate the defiite itegral was Ÿ a f HxL x = FHL - FHaL I this chapter we will use the defiite itegral to solve certai applicatios prolems. I order to otai the defiite itegral, we will first cosider approximatig a solutio, which will lead us to a Riema sum. We will ed up takig a limit of the Riema sums as Ø, which will give us our defiite itegral. Volumes y Cross-Sectios A cross-sectio of a solid S is the plae regio formed y itersectig S with a plae. To calculate the volume of a solid we will use the fact that we ca express volume as follows: Volume = area μ height = Aÿh DEFINTION Volume The volume of a solid of kow itegrale cross-sectioal area AHxL from x = a to x = is the itegral of A from a to, V = Ÿ a AHxL x
2 2 Lecture_06_01. The solid lies etwee plaes perpedicular to the x-axis at x = 0 ad x = 4. The cross-sectios perpedicular to the x-axis etwee these plaes ru from the paraola y =- x to the paraola y = x. The cross sectios are squares with ases i the xy-plae. (See p. 434, Exercise 2.) Calculatig the Volume of a Solid 1. Sketch the solid ad a typical cross-sectio. 2. Fid a formula for AHxL, the area of a typical cross-sectio. 3. Fid the limits of itegratio. 4. Itegrate AHxL usig the Fudametal Theorem. Solids of Revolutio: The Disk Method Cosider the regio ouded y the fuctio y = f HxL, the x-axis, ad the vertical lies x = a ad x =. If we rotate this regio aout the x-axis, what is the volume of the solid otaied?
3 Lecture_06_01. 3 To fid the volume of this solid, we must divide the solid up ito approximatig disks. Cosider the ouded regio. Divide the D ito equal suitervals of legth D x = - a. Let x 0 = a, x 1, x 2,..., x = * e the edpoits of the suitervals ad choose sample poits c 1, c 2,..., c i these suitervals, so that x k lies i the kth k-1, x k D. Stack the disks horizotally ito the regio. The ceter of each disk is o the x-axis. y a x Cosider the kth disk (it is placed i the ith i-1, x i D). The radius of the disk is give y r = y = f Ix k * M ad the height of the disk is its thickess D x. Thus the volume of the disk is V i =pr 2 h =p@ f Ix k * MD 2 D x.
4 4 Lecture_06_01. The total volume is approximated y the sum f Ix * k MD 2 D x. We ca otai etter ad etter aswers if we let get larger ad larger. I other words, i=1 V = lim Ø f Ix * k MD 2 D x = Ÿ p@ f HxLD 2 x a i=1 V = Ÿ a AHxL x = Ÿ a p@rhxld 2 x Fid the volume of the solid that is created whe we rotate the regio ouded y f HxL = x 2, the x-axis, ad the lie x = 1 aout the x-axis. Out[354]= If we cosider the solid otaied y rotatig the regio ouded y the fuctio x = ghyl, the y-axis, ad the horizotal lies y = c ad y = d aout the y-axis, the our formula ecomes V = lim Ø k LD 2 D y = Ÿ dp@ghyld 2 y c i=1
5 Lecture_06_01. 5 Fid the volume of the solid that is created whe we rotate the regio i the first quadrat ouded y f HxL = x 2, the y-axis, ad the lie y = 1 aout the y-axis. Out[332]= Solids of Revolutio: The Washer Method Now cosider the case where the solid has a "hole" or "gap" i it. Suppose our regio is ouded aove y y = f HxL ad elow y = ghxl o the iterval D. To hadle this prolem, we must fid the volume of the outer solid ad sutract the volume of the ier solid.
6 6 Lecture_06_01. Out[384]= The volume would e give y Ÿ a f HxLD 2 x - Ÿ a 2 x = Ÿ a p I@ f HxLD 2 2 M x V = Ÿ a AHxL x = Ÿ a p I@RHxLD 2 2 M x Fid the volume of the solid that is created whe we rotate the regio ouded aove y f HxL = 1 2 the vertical lies x = 0 ad x = 2 aout the x-axis. + x2, elow y ghxl = x, ad
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