Test 2 8., 8.2, 8.4 (density only), 8.5 (work only), 9., 9.2 nd 9.3 relted test mteril nd mteril from prior clsses Locl to Globl Perspectives Anlyze smll pieces to understnd the big picture. Exmples: numericl integrtion vi rectngles re between two curves vi rectngles volume by cylindricl disk or rectngulr box slices totl work vi the work for ech slice = force for ech slice displcement of tht slice series diverges when sequence terms do not get smller. (when they do get smller nything cn hppen)
8. Are nd Volume (Slice nd Conquer) Are by slicing into rectngles with known length Volume by slicing into regions we know the re of Riemnn sums with x or y b dx or b dy π( 2 5 y i) 2 y 5( 2 5 y)2 dy 0 Wht I wnt you to show me... picture, slice, Riemnn sum, integrl
8. Are Steps Sketch grph of the functions to find the enclosed region 2 Sketch picture of Riemnn slice on your grph. 3 Bse of the rectngle? Circle: x or y 4 Which function is lrger in tht vrible (top for x, right for y)? 5 Wht is the height of the rectngle (top-bottom or right-left)? 6 Wht is the Riemnn sum pproximtion? height bse = 7 Wht is nd b (lgebr finds the intersection points)? 8 Write the integrl?
8.2 Volume Steps Sketch grph of the object you wnt to find the volume of 2 Sketch picture of Riemnn slice on your grph 3 Wht shpe is it? Circle: box (length width height) or cylinder/disk (π rdius 2 height) 4 Infinitesiml prt of the slice? Circle: x or y or h 5 Sketch digrm nd show work to solve for ny lengths you need 6 Circle ny we used: Pythgoren theorem or similr tringles 7 Wht is the Riemnn sum pproximtion? 8 Wht is nd b? 9 Write the integrl?
8.2 Volume (Revolutions) nd Arc Length Volume by revolving region bout n xis. Slice. Riemnn sums with x or y b dx or b dy π( 2 5 y i) 2 y 0 5( 2 5 y)2 dy Common forms: b b πr 2 dx nd π(r 2 outer r 2 inner )dx = b πr 2 outer dx b Key is to figure out the rdius (or rdii) vi pics πr 2 inner dx Wht I wnt you to show me... resoning for rdius, integrl
8.2 Volume (Revolutions) nd Arc Length Pythgoren looks good until we see BOTH x, y y x slope = f (x), so y f (x) x rc length x 2 + (f (x) x) 2 = + (f (x)) 2 x b rc length = + (f (x)) 2 dx Wht I wnt you to show me... f, integrl
8.4 Vrying Density Density over length, re or volume, chnging only in dimension (Clc III for others) Substnce mss with density vrying over volume g/cm 3 Popultion quntity like people/squre mile, bcteri/cc. Slice so density is pproximtely constnt: If δ = f (x), then slice x, for constnt density slices mss= density length x, re, or volume δ(x) dx If δ = f (r), then slice from center outwrd, rings popultion= density re b 2πrδ(r) dr Figure out slicing vrible, then δ length or n δ 8./8.2 re or volume in tht vrible
8.5 Work: Vrying Force Work is force distnce displced only pplies if the force is constnt while it is exerted over the distnce Integrls pply when we vry the force. The ide is to slice so tht the force is pproximtely constnt on slice for its displcement. Then like Hook s Lw to stretch (nd hold) spring, where F(x) = kx constnt for displcement x nd W = F(x)dx Sometimes need to clculte the force, like when it is column of wter: mss = density volume, F = mss g Often we won t need to multiply by g like when we hve density tht lredy hs force component: weight (force in lbs) = volume of slice 62.4 lbs/ft 3 work on slice = volume 62.4 lbs/ft 3 slice displcement
Slicing for Volume, Density nd Work Prctice Sheet cylindricl disk volume = π rdius of slice 2 y 2 totl cone volume: 5 0 π( 2y 5 )2 dy Similr : rdius of slice y = 2 5 so r = 2y 5 3 density δ(y) of the cone vries with it s height y: mss = 5 0 δ(y) volume = 5 0 δ(y)π( 2y 5 )2 dy 4 Work to pump the wter out if cone filled to height of 4ft. = F d = ( 62.5lb/ft 3 volume ) d ech slice displced = 4 0 (62.5 (π( 2y 5 )2 dy) (5 y)
9. Sequences list of terms s, s 2,...s n,... often rrnged in fixed pttern lgebric, numeric nd grphicl representtions new vocb: monotone, lternting, recursive, bounded lim s n? converges or diverges? n
9.2 Series: Geometric rtio between ny two consecutive terms is constnt. sum of the first n terms: ( x n ). Creful of # terms nd x ( x n ) strting index. lim = if x < n x x i = i Exmple: 2 2 i=0 i= 2
9.3 Series: Prtil Sums n nd convergence? [9.3, 9.4, 9.5, chpter 0] n= n th prtil sum: n i where i my not be geometric i= sequence of prtil sums S n converges series does so exmine lim n th prtil sums n Exmple: n(n+) S n = n n+ lim S n = n n=
9.3: Limits nd Linerity for Convergence or Divergence terms not getting smller: lim n 0 or DNE, then prtil n sums diverge nd so does the series. Exmple: Linerity: n converges to S nd n= nd k is ny constnt, then n= 5+n 2n+ b n converges to T, n= k n + b n converges to n= ks + T. Appliction : dd two geometric series (converge) Appliction 2: dd divergent & convergent series (diverge) Exmple: convergent n= 2 n= n + ( ) n. If convergent, then subtrct n 2 nd the result should converge.
9.3: Integrl Test Bounds If series hs terms tht re decresing nd positive, the integrl test not only tells us bout convergence, but lso bounds the series: =f() 2 3 4 2 3 4 5 f (x)dx n + f (x)dx
n= n= 9.2 Geometric Series versus 9.3 p-series rtio between ny two consecutive terms is constnt. sum of the first n terms: ( x n ). Creful of # terms nd x ( x n ) strting index. lim if x < n = x x converges if p > nd diverges if p. np n= () n = 2 2 + 4 + 8... geo series, x =.5 < conv to.5.5 n 2 = + 4 + 9... p series: p = 2 > conv by integrl test: terms dec +: x 2 dx = lim b b x 2 dx = lim b x n= n 2 + first term = + b = 0
Is this geometric series? yes no Geometric Series: x i where x is the common rtio nd is constnt. x i = i=0 x i=0 n i=0 x i = ( x n+ ). x provided x <. 2 Cn we pply the Terms not Getting Smller? yes no Terms not Getting Smller: For n, if the lim n 0, n > then the infinite series does not converge. 3 Are the terms decresing nd positive eventully, nd if so is this n integrl we cn do? yes no Integrl Test: For n, if the terms re decresing nd n > 0, then the series behves the sme wy s n dn, & f (x)dx n st term + f (x)dx.
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