Lengths of Plane Curves in Calculus I Lectures
Explore the concept of finding lengths of plane curves using Riemann sums and the Mean Value Theorem in Calculus I lectures. Learn how to calculate the length of curves represented by functions with examples and solutions provided.
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Calculus I Lecture #14 Civil Engineering Department College of Engineering Mustansiriayah University March 2020
Lengths of Plane Curves Lengths of Plane Curves Length of a Curve y=f(x) Suppose the curve whose length we want to find is the graph of the function y = (x) from x = a to x = b. We connect successive points Pk-1and Pkwith straight-line segments that, taken together, form a polygonal path whose length approximates the length of the curve (Figure a). a b Calculus I, Lecture # 14 2 19/04/2025
The line segment in the path has the length given below: The line segment in the path has the length given below: 2+ ?? 2 ??= ?? So the length of the curve is approximated by the sum: ? ? 2+ ?? 2 ??= ?? ?=1 ?=1 In order to evaluate this limit, we use the Mean Value Theorem, which tells us that there is a point ck, with ?? 1 ?? ??, such that: ??= ? ??. ?? Substituting this for yk, the sums for the length of the curve becomes: ? ? ? 2+ ? ??. ?? 2= 1 + ? ?? 2 ?? ??= ?? ?=1 ?=1 ?=1 3 Calculus I, Lecture # 14 19/04/2025
If the function derivative is continuous on [a, b], the limit of the Riemann sum on the right-hand side of the abovementioned Equation has the following value: ? ? ? 1 + ? ?? 2 ??= 1 + ? ? 2 ?? lim ? ??= lim ? ? ?=1 ?=1 Example 1: Find the length of the curve shown in Figure below, which is the graph of the function ? =4 2 ? 3 2 1, 0 ? 1 3 Solution: limits of integration: a=0, b=1 ? =4 2 ?? ??=4 2 .3 2? ? 1 2= 2 2.? 3 2 1, 1 2 3 3 4 Calculus I, Lecture # 14 19/04/2025
Example Example 1 1 ? =4 2 ?? ??=4 2 .3 2? ? 1 2= 2 2.? 3 2 1, 1 2 3 3 2 ?? ?? 2= 8? = 2 2.? 1 2 The length of the curve over x=0 to x=1 is: 2 1 1 ?? ?? ? = 1 + ?? = 1 + 8? ?? 0 0 =2 3.1 1=13 3 2 1 + 8? 8 6 0 5 Calculus I, Lecture # 14 19/04/2025
Example Example 2 2: Find the length of the graph of: : Find the length of the graph of: ? ? =?3 12+1 ?, 1 ? 4 Solution: A graph of the function is shown in Figure below. ? ? =?2 4 1 ?2 Therefore, 2 2 ?2 4 1 = 1 +?4 ?? ?? 16 1 2+1 1 + = 1 + ?2 ?4 2 =?4 ?2 4+1 16+1 2+1 ?4= ?2 The length of the graph over [1, 4] is: 2 2 4 4 4?2 4 ?2 4+1 ?? ?? 4+1 ?3 12 1 64 12 1 12 1 =72 1 ? = 1 + ?? = ?? = ?2??= = 12= 6 ?2 ?1 4 1 1 1 6 Calculus I, Lecture # 14 19/04/2025
Dealing with Discontinuous in Dealing with Discontinuous in dy dy /dx /dx Even if the derivative dy/dx does not exist at some point on a curve, it is possible that dx/dy could exist. This can happen, for example, when a curve has a vertical tangent. In this case, we may be able to find the curve s length by expressing x as a function of y and applying the following analogue of length Equation: 2 ? ? ?? ?? ? (? 2 ?? ? = 1 + ?? = 1 + ? ? Example 3: Find the length of the curve ? = (?/2 2/3from x = 0 to x = 2 . Solution: 1 3 1 3 1 ?? ??=2 ? 2 =1 2 ? 3 2 3 The derivative is not defined at x = 0, so we cannot find the curve s length with the standard equation. 7 Calculus I, Lecture # 14 19/04/2025
We need to express x in terms of y: 3 2=? 2 3 ? 2 ? = 2? 3 2 ? = ? 2 From this we see that the curve whose length we want is also the graph of ? = 2?3/2from y = 0 to y = 1. ?? ??= 2 3 2 1 2= 3? 1 2 ? 2 2 ? 1 1 ?? ?? 1 2 ? = 1 + ?? = 1 + 3? ?? = 1 + 9? ?? ? 0 0 1 1 9.2 2 3 2 = 31 + 9? = 10 10 1 2.27 27 0 8 Calculus I, Lecture # 14 19/04/2025
Lengths of Parametrically Defined Curves Lengths of Parametrically Defined Curves Let C be a curve given parametrically by the equations: ? = ? ? and ? = ? ? , ? ? ? 2+ ?? 2 ??= ?? 2+ ? ?? ? ?? 1 2 = ? ?? ? ?? 1 ??= ? ?? ? ?? 1 = ? ?? ??, ??= ? ?? ? ?? 1 = ? ?? ?? 2 2 ? ?? ?? ?? ?? ? ? = + ?? ? ? 2+ ? ? 2 ?? ? = ? ? 9 Calculus I, Lecture # 14 19/04/2025
Example Example 4 4: : Using the definition, Using the definition, find the length of the circle of radius r find the length of the circle of radius r ( (circumference) circumference) defined parametrically by: defined parametrically by: ? = ?cos? and ? = ?sin?, 0 ? 2? Solution: 2 2 2? ?? ?? ?? ?? ? = + ?? 0 ?? ??= ?sin?, ?? ??= ?cos? 2 2 ?? ?? ?? ?? = ?2sin2? + cos2? = ?2 + 2? 2?= 2?? ?2 ?? = ? ?0 ? = 0 10 Calculus I, Lecture # 14 19/04/2025
Rotation about a Vertical Axis Rotation about a Vertical Axis Example 5: Find the length of the asteroid given in the Figure below, and defined parametrically by the following equations: ? = cos3? and ? = sin3?, 0 ? 2? Solution: Because of the curve s symmetry with respect to the coordinate axes, its length is four times the length of the first-quadrant portion. 2 = 3cos2? sin? ?? ?? 2= 9cos4?.sin2? 2 ?? ?? = 3sin2? cos? 2= 9.sin4?.cos2? 2 2 ?? ?? ?? ?? 9cos2?.sin2? cos2? + sin2? + = ???????? ??? 0 ? ? 9cos2?.sin2? = 3 cos?.sin? = = 3cos?.sin? 2 11 Calculus I, Lecture # 14 19/04/2025
Length of the first-quadrant piece is: ? 2 3cos?.sin? ?? =3 ? 2=3 sin2?0 ? = 2 2 0 The length of the asteroid is four times the last answer, that is ? = 4 3 2= 6 Homework Find the lengths of the following curves: A. ? = 1 ? and ? = 2 + 3?, 2/3 ? 1 B. ? = cos? and ? = ? + sin?, 0 ? ? C. ? = ?3 ? = 2?2/2, and 0 ? 3 3 2/3, D. ? = ?2/2 and ? = 2? + 1 0 ? 4 E. ? = 8cos? + 8?sin? and ? = 8sin? 8?cos?, 0 ? ?/2 12 Calculus I, Lecture # 14 19/04/2025
