MTH 122 Fall 2008 Essex County College Division of Mathematics Handout Version 10 1 October 14, 2008


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1 MTH 22 Fll 28 Essex County College Division of Mthemtics Hndout Version October 4, 28 Arc Length Everyone should be fmilir with the distnce formul tht ws introduced in elementry lgebr. It is bsic formul for the liner distnce between two points in the plne. It sttes tht the distnce between (x, y ) nd (x 2, y 2 ) is d = (x 2 x ) 2 + (y 2 y ) 2. This distnce, of course, is for line connecting those two points. However, wht if we hve curve nd we wnt to know the distnce long tht curve between two points? We will bsiclly cut the curve into n infinite number of smll liner sections, nd then dd these sections together to get the rc length, or distnce between two points on the curve. Here definite integrl cn be used to find the rc length, where we hve curve, f (x), nd two points on this curve tht re connected by curve tht is continuously differentible on the intervl. Arc Length Formul: If f is continuous on [, b], then the length of the curve y = f (x), x b, is + [f (x)] 2 dx. This cn lso be written s + ( ) dy 2 dx. dx You should notice, tht we re doing these integrtions with respect to x, but you my recll from erlier problems, tht it is sometimes esier to integrte with respect to y. Arc Length Formul: If g is continuous on [c, d], then the length of the curve x = g (y), c y d, is d + [g (y)] 2 dy. This cn lso be written s c d c + ( ) dx 2 dy. dy Ech of these formuls will be discussed in clss, nd I will minly try to relte wht we know bout locl linerity nd integrtion to derive these formuls. The book, of course, proves these formuls nd you my be interested in reding the textbook for more precise explntion. This document ws prepred by Ron Bnnon using L A TEX 2ε.
2 . Exmple. Set up n integrl to compute the length of the curve y = x 3 from x = to x = 5. Work: Here I will be using this formul. + ( ) dy 2 dx. dx Plugging in, I get. = = (3x 2 ) 2 dx + (3x 2 ) 2 dx + 9x 4 dx You should notice tht you were not sked to evlute this integrl. 2 ctully integrte this, but it s confusing in its exct form. Mthemtic cn 2 Are of Surfce of Revolution Definition: If the grph of continuous function is revolved bout line, the resulting surfce is surfce of revolution. Here we will let y = f (x), where f hs continuous derivtive on the intervl [, b]. The re S of the surfce of revolution formed by revolving the grph of f bout horizontl or verticl xis is S = 2π r (x) + [f (x)] 2 dx. Here y is function of x. where r is the distnce between the grph of f nd the xis of revolution. On the other hnd, if x = g (y) on the intervl [c, d], then the surfce re is S = 2π r (y) + [g (x)] 2 dy. Here x is function of y. where r is the distnce between the grph of g nd the xis of revolution. Ech of these formuls will be discussed in clss, nd I will minly try to relte wht we know bout locl linerity nd integrtion to derive these formuls. The book, of course, proves these formuls nd you my be interested in reding the textbook for more precise explntion. 2 Approximtely 25.68, which is slightly longer thn the liner distnce between those points on y = x 3. 2
3 2. Exmple. Find the re of the surfce formed by revolving the grph of on the intervl [, ] bout the xxis. Work: Here I m using the formul S = 2π Mking the substitutions, I get. S = 2π = 2π Using simple usubstitution, it follows. f (x) = x 3 r (x) + [f (x)] 2 dx. x 3 + [3x 2 ] 2 dx x 3 + 9x 4 dx 2π x 3 + 9x 4 dx = π 8 u /2 du ] = πu u 27 = π ( ) Exmples. Find the re of the surfce generted by rotting the curve y = e x, x, bout the x xis. 3 3 This is not esy, but I think you ll eventully get h S = π e p + e 2 + ln e + p + e 2 2 = ln + i 2. 3
4 2. Show tht the length of rc of the grph on the intervl [/2, 2] is 33/6. y = x x 3. Show tht the length of rc of the grph on the intervl [, 8] is ( 4 3/2 4 3/2) /27. (y ) 3 = x 2 4
5 4. Show tht the surfce formed by revolving y = x 2 on the intervl [, 2 ] bout the yxis is 3π/3. 5. The solid formed by revolving the region between /x nd the xxis, for x is clled Gbriel s Horn. Wht is most weird bout this object is tht it cn be shown to hve finite volume, but its surfce re is infinite. Setup nd evlute the integrl for its surfce re nd show tht this integrl is divergent. 5
6 6. Find the totl length of the grph of the stoid 3 x y 2 = 4. A grph is provided s guide Figure : Complete grph of 3 x y 2 = 4. 6
We divide the interval [a, b] into subintervals of equal length x = b a n
Arc Length Given curve C defined by function f(x), we wnt to find the length of this curve between nd b. We do this by using process similr to wht we did in defining the Riemnn Sum of definite integrl:
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