Slide 1. Slide 2. Slide 3. Solids of Rotation:

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1 Slide 1 Solids of Rotatio: The Eggplat Experiece Suz Atik Palo Alto High School Palo Alto, Ca EdD; NBCT, AYA Math satik@pausd.org May thaks to my colleague, Kathy Weiss, NBCT, AYA Math, who origially desiged this activity ad is happy to share it with others! Slide Materials Needed: Groups of two to four studets Oe eggplat per group (approximately the same shape ad weight) A plastic serrated kife for each group A cetimeter ruler for each group Ulied paper for each group Cetimeter graph paper for each group Newspaper to keep it easy to clea up Slide Simple Directios Each group has a eggplat, which may be shaped differetly tha aother group s, but which has approximately the same weight. Your task is to devise a method usig Calculus techiques* to approximate the volume of your eggplat (cm ). Your method should ot ivolve water, should be essetially completed i class today, ad should ivolve fewer tha 15 pieces of eggplat. You will graph a cross-sectio of your eggplat, labelig importat poits. Usig those poits, as a class we ll review how to fid a curve of best fit, which you will do for your ow data, ad will use your curve for the remaider of this applicatio. *Slicig the eggplat with cuts perpedicular to the axis of symmetry

2 Slide 4 Team Time: 10 to 0 mi s (Teacher Team Time: about 5 miutes) The directios will be re-projected i a momet. As a team ( to 4 studets), braistorm what you will do cosider the implicatios! Execute your pla! Slide 5 Simple Directios Each group has a eggplat, which may be shaped differetly tha aother group s, but which has approximately the same weight. Your task is to devise a method usig Calculus techiques* to approximate the volume of your eggplat (cm ). Your method should ot ivolve water, should be essetially completed i class today, ad should ivolve fewer tha 15 pieces of eggplat. You will graph a cross-sectio of your eggplat, labelig importat poits. Usig those poits, as a class we ll review how to fid a curve of best fit, which you will do for your ow data, ad will use your curve for the remaider of this applicatio. *Slicig the eggplat with cuts perpedicular to the axis of symmetry Slide 6 10 slices are less tha 15 What oe group did First, the cross sectio o a set of axes Measure off legths of cm o the x-axis (coveiet)

3 Slide 7 They foud the heights at each cm icremet, ad idetified a geeral slice callig it the k th slice The K th Slice Slide 8 They pulled out the k th slice, ad looked at its dimesios. They oticed it was a cylider, or pretty close to oe if they had made more slices it would have bee closer to a ideal cylider (or disk). The Kth Slice is a cylider r k = f ( x * k ) Notice: They chose x k* as the right edpoit of the partitio. h k =, because they used a coveiet set of regular partitios. rk hk f ( xk ) h r h V Kth = k Slide 9 To approximate the volume, they just added up the volume of each of the disks. They oticed the commo factors of, from the formula, as well as the commo factor,, the width of the slice, aka, the height of the 10 cylider/disk V Eggplat rk hk (.1) () (4.1) () (0.6) () or Dx (11.5) cm

4 Slide 10 to this poit This project works for Geometry ad Algebra studets. Bous: It foreshadows the skills they ll use i Calculus! This activity ca be repeated i successive years successfully. Add the ext level of complexity each year Slide 11 move the studets ito Pre-Calculus work.. Fid a curve of best fit usig a statistical egie: We ll use the TI-89, but other tools are available as well. Put the x-coordiates ito List1 ad the y- coordiates ito List Press F4: Calc; #: Regressios; You have a variety of regressio choices: look at the possible polyomials: Liear, Quadratic, Cubic, ad Quartic Slide 1 Cosideratios Whe summig the best fits for the regressio you will use formulas beyod k ad k Although you ca use the additioal summatio formulas up to k 8 I advise agaist it. It is too tedious for the average HS studet. Let them use the powerful calculators, or calculus.

5 Slide 1 for this group The liear regressio (silly to eve cosider) yielded the curve y = -.05x , stored i y1(x), with a correlatio coefficiet of r = (terrible!) ad a coefficiet of determiatio of r =.004, a predictably bad fit. Slide 14 The The quadratic regressio yielded the curve (stored i y(x)) y = x x with a coefficiet of determiatio of r = , a much better fit, although it did t break the 0.9 threshold. Slide 15 followed by The cubic regressio yielded the curve (stored i y(x)) y =.00059x x x with a coefficiet of determiatio of r =.9566, a solidly good fit!

6 Slide 16 ad, fially, The quartic regressio yielded the curve (stored i y4(x)) y = x x x x with a coefficiet of determiatio of r = , a excellet fit, our best fit. Slide 17 Cruchig time! Oto summatio, aka, a Riema Sum The height of the cylider will be the legth of the iterval, 0, divided by, the umber of partitios. All of the heights will be the same Agai, let the radius of the disk be take from the right side of the slice. The height will be the y-value at the associated x- value. The x-value depeds o which partitio you re o from the begiig poit. Pull out the k th disk to wrap your mid aroud it. Geeralize it! Istead of 10 cogruet partitios, cosider that there are cogruet partitios. First, otice x k = 0 + k(0/) The, r k = f(x k ) = f(0 + k(0/)) ad h k = 0/, for as may partitios () as you d like. Slide 18 Establish the Volume of the k th Slice V(kth slice) = r h 0k 0 f 0 0k 0 f We ll use the regressio that was the BEST fit for f(x) the quartic w/r =.996

7 Slide 19 Write the approximate volume of the eggplat of a sum of all of the slices for ay value of. Let V = the approximate volume of the eggplat give partitios 0k 0 Now V y4 PreCalculus studets ca or expad this sum 0 0 for ay value of k V y4, or for geerally with a calculator Slide 0 Movig toward Calculus The expasio of 0 0k V y4 is, i abbreviated decimal format, ( ) V 8 V 10 = cm very close to the cm from the first summatio V 5 = cm Studets will otice the similarity betwee the leadig coefficiet for V ad the volume approximatios. Slide 1 What if. the umber of partitios,, is ulimited?, ot a umber?, ifiite???!!!! Calculus emerges as studets eter this expressio ito their calculators 0 0k V Lim y4 V = cm The volume is o loger approximated, ad we build toward the itegral!

8 Slide The Itegral is a Natural Extesio of the Riema Sum V Of course, both values result i a volume of 764. cm, approximately. Lim 0 0k y4 1 k V x0 y 4 ( x) x0 dx Slide It s memorable I their exit iterviews, the studets reported that this activity is memorable ad is meaig-makig. It s the Hads o K th slice Modelig from data Coectios to HUGE cocepts Group talk, isights, ad work Appropriate use of techology Slide 4 Solids of Rotatio? Cite the eggplat experiece The atural symmetry of a eggplat let itself to the idea of a solid created by rotatig a cotiuous fuctio about a lie (like the x-axis). Move your class to the geeral solid of rotatio. Begi with a positive fuctio o [a, b] ad rotate it about the x-axis.

9 Slide 5 A Solid of Rotatio x = a f(x) is cts over [a, b] x = b K th slice is just a slice of eggplat! Rotate about the X-axis Slide 6 Pull out the K th Slice ad idetify its dimesios The K th Slice V k = r h V k = (f(x k)) Dx X-axis Radius (the height up to the curve) is the y value, aka, f(x k) for each slice ( of the eggplat ). The total volume is the volume of all of these slices as Width, aka, we icrease the umber of slices, ad decrease the Height, is Dx, i size of Dx. this example The Riema Sum: V = Lim f ( xk ) Dx Dx 0 b a xb where Dx = or V = f ( x) dx xa Slide 7 Some getle guided practice Graph the followig fuctios i the idicated domai Sketch the rotatio uder the curve about the idicated axis Pull out the k th slice i the solid ad fid its dimesios Write the expressio for the volume of the k th slice Write the Riema sum for the volume Use itegratio to fid the exact volume 1. f(x) = x x o x i [0, ] about the x-axis. (Sphere). f(x) = si(x) o x i [0, /] about the y-axis. (Washers)

10 k Slide 8 Getle Guided Practice Solutios 1. f ( x) x x o [0, ] rotated about. f ( x) si xo 0, rotated about the x-axis. the y-axis. Is it a circle? => y = x x => The K th Slice is a washer. y = - (x x + 1) + 1 => (x 1) + y = 1 r ier = x k = si 1 (y k) C = (1, 0) ad r = 1 r outer =, a costat Note: So this is a sphere of radius 1. We ll expect the aswer to be 4 / uits. r k = y k = x xk h k = Dx V k = r k h k = (x k x k ) Dx V = V = Lim (xk k 1 (x x ) dx x0 V = ( x x ) x x0 xk ) Dx (4 8 ) 4 h k = Dy V k th Washer = V(outer disk) V(ier disk) 1 V K = Dy (si ( yk )) Dy 1 V K = [si ( y k )] Dy 4 1 V = Lim (si ( yk )) Dy V = (si ( y)) dy y0 4 V = uits Slide 9 Thaks so much! Time for some Eggplat parmesa!