Length of a Plane Curve (Arc Length)

Transcription

1 Length of a Plane Curve (Arc Length) SUGGESTED REFERENCE MATERIAL: As you work through the problems liste below, you shoul reference Chapter 6. of the recommene textbook (or the equivalent chapter in your alternative textbook/online resource) an your lecture notes. EXPECTED SKILLS: Be able to fin the arc length of a smooth curve in the plane escribe as a function of x or as a function of y. PRACTICE PROBLEMS: For problems -3, compute the exact arc length of the curve over the given interval.. y = x 3 from x = to x = y = x ln(x) 6 + ln for x 3. y = 3 (x ) 3/ for x Consier the curve efine by y = x for x. (a) Compute the arc length on the interval [, t] for t <. (Your arc length will epen on t.) ( ) t sin (b) Use your answer from part (a) to compute the arc length on the interval [, ]. (Hint: You will nee to introuce a limit.)

2 (c) Confirm your answer from part (b) by using geometry. On the interval [, ], the curve is of a circle with a raius of. So, the length shoul be of the circumference; that is, Length = r= r = () =. 5. Consier F (x) = x t t. Compute the arc length on [, 3] 6. Consier the curve efine by f(x) = ln x on [, e 3] (a) Set up but o not evaluate an integral which represents the length of the curve by integrating with respect to x. L = e 3 + x (b) Set up but o not evaluate an integral which represents the length of the curve by integrating with respect to y. 3 L = + e y y 7. Consier the curve efine by f(x) = tan x on [ 3, ] (a) Set up but o not evaluate an integral which represents the length of the curve by integrating with respect to x. L = 3 + sec x (b) Set up but o not evaluate an integral which represents the length of the curve by integrating with respect to y. L = + ( + y ) y 3. Consier the curve efine by y = sin x for x. (a) Set up but o not evaluate an integral which represents the length of the curve. + cos x

3 int sqrt Kx, x arcsin x with Stuent Calculus AntierivativePlot, AntierivativeTutor, ApproximateInt, ApproximateIntTutor, ArcLength, ArcLengthTutor, Asymptotes, Clear, CriticalPoints, CurveAnalysisTutor, DerivativePlot, (b) Estimate the value of your integral from part (a) by using a Mipoint Approximation with three rectangles DerivativeTutor, DiffTutor, ExtremePoints, FunctionAverage, FunctionAverageTutor, FunctionChart, offunctionplot, equal GetMessage, with. GetNumProblems, GetProblem, Hint, InflectionPoints, IntTutor, Integran, InversePlot, InverseTutor, LimitTutor, Below is the graph of y = MeanValueTheorem, MeanValueTheoremTutor, + cos NewtonQuotient, NewtonsMetho, NewtonsMethoTutor, PointInterpolation, x onriemannsum, the interval RollesTheorem, Roots, [, Rule, ] Show, along with three ShowIncomplete, ShowSolution, ShowSteps, Summan, SurfaceOfRevolution, rectangles of equal with SurfaceOfRevolutionTutor, whose heights Tangent, TangentSecantTutor, were etermine TangentTutor, by the function value TaylorApproximation, TaylorApproximationTutor, Unerstan, Uno, VolumeOfRevolution, at the mipoint of each resulting subinterval. VolumeOfRevolutionTutor, WhatProblem RiemannSum sqrt C cos x, x =..Pi, metho = mipoint, partition =3,output = plot () () Using these rectangles, K. K A mipoint Riemann sum approximation of f x, where f x = Ccos x an the partition is uniform. The approximate value of the integral is Number of subintervals use: 3. x 5 + cos x ( + ) 7 9. Recall the efinitions of Hyperbolic Sine & Hyperbolic Cosine from Math : Hyperbolic Sine sinh x = ex e x Hyperbolic Cosine cosh x = ex + e x The sketches of y = cosh x an y = sinh x are shown below. The ashe curves are calle Curvilinear Asymptotes, which escribe the en behavior of the functions. 3

4 (a) Show that cosh x sinh x = cosh x sinh x = (cosh x + sinh x)(cosh x sinh x) ( ) ( e x + e x = + ex e x e x + e x = (e x )(e x ) = ) ex e x (b) Verify that f(x) = sinh x is an o function. satisfies the ientity f( x) = f(x).) (Hint: Recall an o function To verify that a function is off, we check that f( x) = f(x). We compute by appealing to the efinition of sinh x from above. Thus, f(x) = sinh x is o. sinh ( x) = e x e ( x) = e x e x ( ) e x e x = = sinh x

5 (c) Show that (sinh x) = cosh x an euce that cosh x = sinh x + C. Thus, as a result, (sinh x) = = ( ) e x e x ( ex e x = ex + e x = ex + e x = cosh x cosh x = sinh x + C. (We coul have also verifie this integration formula by integrating the given efinition of cosh x.) () Show that (cosh x) = sinh x an euce that sinh x = cosh x + C. Thus, as a result, (cosh x) = = ( ) e x + e x ( ex + e x = ex e x = ex e x = sinh x sinh x = cosh x + C. (We coul have also verifie this integration formula by integrating the given efinition of sinh x.) (e) A telephone wire which is supporte only by two telephone poles will sag uner its own weight an form the shape of a catenary as shown below. ) ) 5

6 Consier a telephone wire that is supporte by two poles (one at x = b an the other at x = b), as in the iagram below. ( x The shape of the sagging wire can be moele by y = a cosh, where a > a) an b x b. What is the length of the wire? ( ) b A = a sinh a 6