8.1 Arc Length. What is the length of a curve? How can we approximate it? We could do it following the pattern we ve used before

Transcription

1 8.1 Arc Legth Wht is the legth of curve? How c we pproximte it? We could do it followig the ptter we ve used efore Use sequece of icresigly short segmets to pproximte the curve: As the segmets get smller (d the umer of segmets gets lrger,) it seems resole tht the pproximtio will get closer to the ctul legth. But how log is ech segmet? By the Me Vlue Theorem, we kow tht, for every itervl, there exists some xi * such tht f(xi) f(xi-1) = f (xi * ) (xi xi-1) or yi = f (xi * ) x so

2 Pi-1 Pi = ( x) 2 + ( y i ) 2 = ( x) 2 + [f (x i ) x] 2 = 1 + [f (x i )] 2 x So L = 1 + [f (x)] 2 dx Ex: Fid the legth of y 2 = x 3 etwee the poits (1, 1) d (4, 8). Wht does the grph look like? y = x 3/2 dy dx = 3 2 x1/2 so ( dy dx )2 = 9 4 x so 4 L = x 4 dx

3 Sometimes, isted of simply mesurig prticulr rc, we wt to set up rc legth fuctio tht will give us formul for the legth i geerl. Ex 4: Fid the rc legth fuctio for the curve y = x 2 1/8 lx, with P0 (1,1) s the strtig poit. Fid f (x)= The fid 1 + [f (x)] 2 = So 1 + [f (x)] 2 = Which mes tht the rc legth, s(x), is give y x 1 + [f (x)] 2 1 = Wht s the rc legth log the curve from (1, 1) to (2, f(2))?

4 8.2 Are of Surfce of Revolutio Wht is surfce? Lterl surfce of circulr cylider: Lterl surfce re of cylider: A = 2πrh Lterl surfce of coe:

5 Lterl surfce re of coe: A = πrl (from erlier result) Surfces of Revolutio If the surfce is more complicted th coe or cylider, how c we clculte its surfce re? First, imge the surfce creted whe you revolve curve roud the x-xis: The, imgie rekig the curve dow ito smller pieces, which re revolved roud the x-xis to mke ds. Wht is the re of this d?

6 The trick is to look t tht d s eig simply piece of right, circulr coe. O pge 546, Stewrt explis why the surfce re of this d is A = 2πrl where l is the slt legth (or width) of the d. So the surfce hs re S = 2πrl dx S = 2πf(x)(rc legth) dx S = 2πf(x) 1 + [f (x)] 2 dx Ex: The rc of the prol y = x 2 from (1, 1) to (2, 4) is rotted out the y-xis. Fid the re of the resultig surfce. The ook gives 2 solutios oe i x d oe i y. We ll go through the first oe.

7 S = 2πx ds 2 1 S = 2πx 1 + ( dy dx )2 dx y = x 2 dy dx = 2x 2 S = 2πx 1 + 4x 2 dx 1 We c use u sustitutio to evlute the itegrl: u = du =

8 (Look t Pge 549 for the other versio of the solutio.) 8.3 Applictios Physics d Teeter-totters Questio: If you hve dult d child who wt to ply o teeter-totter, how should they sit if the dult is much hevier th the child?

9 I fct, Archimedes discovered tht the teeter-totter will lce if m1d1 = m2d2 (Lw of the Lever) where d1 d d2 re mesured from the ceter of the teeter-totter (more geerlly clled the fulcrum): Ech of the products midi is mesure of how much ech perso is pushig dow o the ed of their side of the teeter-totter. This is clled the torque, which is oe type of momet. Becuse the force is eig pplied to the ed of lever coected to sigle poit, the torque is rottiol force. If, isted of lie, we hve thi plte of y give shpe, the we c tlk out the ceter of mss s eig the poit where the plte will lce horizotlly, s show:

10 We c lso cosider the ceter of mss for our teeter-totter exmple it would lie t the fulcrum of the teeter-totter. Lie the teeter-totter log the x-xis, where m1 lies t x1 d m2 lies t x2, d lel the ceter of mss s x. The d1 = x - x1 d d2 = x2 - x We kow: m1d1 = m2d2 so the lso d the m1(x - x1) = m2(x2 - x ) m1 x + m2 x = m1x1 + m2x2 which mes tht x = m 1x 1 + m 2 x 2 m 1 + m 2 If you hve system with msses t differet poits log the x-xis, you c exted this result further: x = i=1 m ix i i=1 m i = m ix i i=1 m

11 where m = i=1 m i is the totl mss of the system d M = i=1 m i x i is the momet of the system out the origi which mkes the equtio ove ito mx = M Similrly, we c exted these results to msses rrged o the ple to get My = i=1 m i x i the momet of the system out the y-xis d Mx = i=1 m i y i the momet of the system out the x-xis with the correspodig equtios x = M y m d y = M x m Ex: Fid the momets d ceter of mss of the system of ojects tht hve msses 3, 4, d 8 t the poits (-1, 1), (2, -1), d

12 (3, 2), respectively. We eed to fid My d Mx My = i=1 m i x i = 3(-1) + 4(2) + 8(3) = 29 Mx = i=1 m i y i d the we eed to fid x d y m = so i=1 m i = x = M y m = y = M x m = So, ow tht we hve formul for the ceter of mss of lie segmet, c we exted this to fid the ceter of mss for the thi plte of give shpe (which is lso kow s the cetroid of the regio R)? To mke it esier, ssume the plte hs uiform desity, p. (Hd wvig) As efore, we ll rek the regio dow ito rectgles tht re f(x ) i y x, where the cetroid of the ith rectgle is give y (x i, 1 f(x 2 i)), where x i is the midpoit of the suitervl i the x directio. The mss of ech rectgle is the

13 mi = p f(x ) i x (Desity * Are) d the momet of the ith suitervl out the y-xis is give y p f(x ) i x * x, i which mes tht the momet of the regio s whole out the y-xis is: My = lim i=1 p x i f(x i ) x = p x f(x) dx Similrly, we c fid Mx, which is give y Mx = lim p 1 i=1 [f(x 2 i)] 2 x = p 1 2 [f(x)]2 dx Filly, we c clculte x d y : x = M y y = M x = p x f(x)dx m p f(x)dx = = p 1 2 [f(x)]2 dx m p f(x)dx x f(x)dx f(x)dx = 1 A = 1 2 [f(x)]2 dx f(x)dx x f(x)dx = 1 1 A 2 [f(x)]2 dx Ex #6: Fid the cetroid of the regio ouded y the lie y = x d the prol y = x 2. Step 1: Fid the re of the regio A = (x x 2 ) dx

14 Step 2: Clculte x d y x = 1 A x [f(x) g(x)]dx y = 1 A 1 2 [[f(x)]2 [g(x)] 2 ]dx