1 Using Integration to Find Arc Lengths and Surface Areas


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1 Novembe 9, 8 MAT86 Week Justin Ko Using Integtion to Find Ac Lengths nd Sufce Aes. Ac Length Fomul: If f () is continuous on [, b], then the c length of the cuve = f() on the intevl [, b] is given b s = + (f () d. Remk: We cn emembe this fomul using the diffeentil nottion (ds = (d + (d ds = + ( d d ds = + d The c length s cn be ecoveed b integting the diffeentil, s = ds. ( d d. d Intuition: We cn ppoimte the length of cuve with polgonl pth of line segments of the fom s i = ( + ( i. B the men vlue theoem, thee eists i in the subintevl of length such tht i = f ( i ), so the ppoimtion cn be witten s s i = ( + (f ( i ) ) = + (f ( i )). () If we ptition [, b] into n unifom subintevls nd ppoimte the e with polgonl pth of line segments of the fom (), tking the limit s n implies Ac Length = lim n i= + (f ( i )) = + (f () d s Figue: The length of the line segment cn be ecoveed using the Pthgoen theoem. The length of ech ppoimting line segment is given b ( s = ( + (. Pge of 7
2 Novembe 9, 8 MAT86 Week Justin Ko. Sufce Ae Fomul: If f () is continuous on [, b], then the sufce e of solid of evolution obtined b otting the cuve = f(). Aound the is on the intevl [, b] is given b (povided tht ) S = π + (f () d.. Aound the is on the intevl [, b] is given b (povided tht = f() ) S = πf() + (f () d. Remk: We cn emembe this fomul using the diffeentil nottion S = π ds (is ottion) o S = π ds (is ottion). This sufce e is ecoveed b integting the cicumfeence of cicle with espect to the c length. Intuition: If the sufce it obtined b otting bout the is, then we cn ppoimte the sufce e with ectngul bnd of the fom A i = π i s i = π i + (f ( i )), () whee i is point in subintevl of length. If we ptition [, b] into n unifom subintevls nd ppoimte the e with polgonl pth of line segments of the fom (), tking the limit s n implies n Sufce Ae = lim π i + (f ( i )) = π + (f () d. i= Figue: The e of the hoizontl bnd cn be estimted with ectngle. The height of the ectngle is estimted b the c length s, nd the width of the ectngle is the cicumfeence of cicle with dius Ae = π s. Remk: If the sufce it obtined b otting bout the is, then the onl modifiction is the dius of the cicle is = f() insted. Pge of 7
3 Novembe 9, 8 MAT86 Week Justin Ko.3 Emple Poblems.3. Ac Length Stteg:. (Optionl) Dw the Cuve: Dw the cuve in the (, ) plne.. Set up the definite integl: Use the fomul fo the c length teting the cuve s function of o (depending on which esults in the simple integl). 3. Compute the integl. Poblem. ( ) Show tht the cicumfeence of cicle with dius is π. Solution. B smmet, it suffices to compute the c length of the semicicle = on the domin [, ] nd multipl ou finl nswe b. = Finding the Integl: Since d d =, the c length of the semicicle is given b + ( d d = d + d = d. Computing the Integl: The integnd looks like the deivtive of the sin (), but we need to do some lgebic mnipultion fist. We multipl the numeto nd denominto b to conclude We will use the chnge of vibles u =, d = d. ( ) du d =, du = d = u =, = u =. Unde this chnge of vible, we hve d = ( ) du = u sin (u) u= u= ( π = + π ) = π. Theefoe, the cicumfeence of semicicle is π nd the cicumfeence of the cicle is π. Remk: In genel, c length integls e quite hd to compute becuse of the sque oot tem in the integnd. We will len moe tools such s tigonometic substitution tht will llow us to solve much moe difficult poblems. Poblem. ( ) Find the length of the cuve = ln(sec()) on the domin [, π ]. Pge 3 of 7
4 Novembe 9, 8 MAT86 Week Justin Ko Solution. We hve to use the c length fomul in tems of d. Finding the Integl: Since d sec() tn() d = sec() = tn(), the c length of the cuve is given b + ( d d = + tn () d = sec() d = d if we use the tigonometic identit + tn (θ) = sec θ. sec() d Computing the Integl: This is stndd integl (see Week Section.5. Poblem ),.3. Sufce Ae Stteg: sec() d = ln sec() + tn() = π = = ln( + ).. (Optionl) Dw the Pojected Cuve: Dw pojection of the cuve onto the (, ) plne.. Set up the definite integl: Find fomul fo the sufce e b using the sufce e fomuls. 3. Compute the integl. Poblem. ( ) Show tht the sufce e of sphee with dius is π. Solution. We compute the sufce e in two ws: Rotting ound the is The sphee is obtined b otting the cuve = on the domin [, ] nd ound the is. = Finding the Integl: Since d d =, the sufce e of the sphee is given b ( d π + d = π d + d = π d. Computing the Integl: This integl is es to compute, π d = π = = = π. Rotting ound the is B smmet, it suffices to compute the sufce e of the hlf sphee is obtined b otting the cuve = on the domin [, ] nd ound the is nd multipling ou finl nswe b. Pge of 7
5 Novembe 9, 8 MAT86 Week Justin Ko = Finding the Integl: Since d d =, the sufce e of the sphee is given b π + ( d d = d π + d = π Computing the Integl: We will use the chnge of vibles u =, d. du d = du = d = u =, = u =. Unde this chnge of vibles, we hve π d = π u du = π u u= u= = π ( ) = π. Theefoe, the sufce e of hemisphee is π nd the sufce e of the sphee is π. Pge 5 of 7
6 Novembe 9, 8 MAT86 Week Justin Ko Applictions to Phsics. Wok Definition. Recll tht the wok to move n object fom the point = to = b with vible foce F () is given b F () d. Emple. Thee e sevel phsicl emples whee the fomul fo the foce F () is well known,. Newton s Second Lw of Motion: Let m be the mss of n object nd let be the cceletion of the object, then the foce equied to move the object is F () = m.. Hooke s Lw: Let k > be the sping constnt. If is the distnce fom the sping s equilibium point, then the foce equied to mintin the sping units is F () = k. 3. Gvittionl Foce: Let m nd m be the msses of two objects septed t distnce. The gvittionl foce of ttction (with gvittionl constnt G) is. Wok Requied to Move Fluids.. Filling Tnk F () = Gm m. Fomul: The wok to fill tnk with coss sectionl e A() with stting height = to ending height = b with fluid of densit ρ is given b.. Empting Tnk ρga() d whee g is the cceletion of gvit. Fomul: The wok to empt tnk with coss sectionl e A() with stting height = to ending height = b with fluid of densit ρ into contine of height = h is given b ρg(h )A() d whee g is the cceletion of gvit. Intuition: The wok to move the fluid is ppoimted b the sum of wok equied smll shells of fluid. The wok to move smll shell of fluid is modeled using Newton s lw: Wok = foce distnce = mss cceletion distnce. The mss of shell of fluid with cosssectionl e A() is given b densit volume: mss = ρa(). (see Week Section.) To fill tnk, we need to move this shell of fluid distnce of height, so Wok = lim ρa() g = ρga() d. n= To empt tnk, we need to move this shell of fluid distnce of height h, so Wok = lim ρa() g (h ) = ρg(h )A() d. n= Pge 6 of 7
7 Novembe 9, 8 MAT86 Week Justin Ko.3 Emple Poblems Poblem. ( ) A foce of N is equied to hold sping tht hs been stetched fom ntul length of. m to length of.5 m. How much wok is equied to stetch the sping fom.5 m to. m? Solution. We need to use Hooke s Lw. Finding the Integl: The wok equied to stetch the sping is..5 F () d =..5 k(.) d, since the equilibium point of the sping is =.. To find the sping constnt k, notice tht = k(.5.) = k =.5 = 8. Compute the Integl: This integl is es to compute,..5 8(.) d = 8 =. =.5 = 3 (joules). Poblem. ( ) Conside cicul conicl tnk with height m nd dius m. Suppose tht the wte level is cuentl 8 m high. Given tht the densit of wte is ρ = kg/m 3, find the wok equied to empt the tnk. Solution. We need to use the fomul to empt tnk. Finding the Integl: The wok empt the fluid out of tnk m high is 8 ρg( )A() d. Using simil tingles, the coss sectionl e is disc with dius = = = 5. Theefoe, the coss sectionl e is A() = π = π 5. m Compute the Integl: This integl is es to compute. Using the fct tht g = 9.8 m/s, m 8 8 ρg( )A() d = 9.8 π 5 ( ) d ( = π 3 3 ) =8 = ( = π ) (joules). Pge 7 of 7
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