Calculus Mrs. Cartledge Arc length Phantom’s Revenge Kennywood Park Sec 7.4 Objectives Determine the length of a curve. Area Under a Curve (revisited) We use the definite integral to find the area under a curve. The limits of integration are a and b. The height of the rectangle is represented by f(x), the width by dx and the sum by the definite integral. Sum of Disks as the Volume (revisited) b V [ R ( x )] dx 2 a Finding Arc Length We want to determine the length of the continuous function f(x) on the interval [a,b] . Initially we’ll need to estimate the length of the curve. We’ll do this by dividing the interval up into n equal subintervals each of width x and we’ll denote the point on the curve at each point by Pi. We can approximate the curve by a series of straight lines connecting the points. The length of one segment P5 P4 The length of all segments n L ( xi yi ) 2 2 i 1 n i 1 i 1 i 1 ( xi yi ) * Δxi 2 2 xi n n 1 [ f ( x )]2 dx a ( xi yi ) 2 xi b 2 2 * Δxi yi (1 ) * Δxi 2 xi 2 Definition of Arc Length Example Applying the Definition Find the length of the curve y x for 0 x 1. 2 dy 2 x, this is continous on 0,1. Therefore, dx L 1 2 x using NINT 1 0 L 1.479 2 Example Find the length of the specified curve. = .881374 Algorithm to Find Arc Length 1. Determine whether the length is with respect to x or y, and then find the endpoints for the interval. 2. Find y’(x) or x’(y). 3. Plug the derivative into the formula: 4. Evaluate. Sample Find the length of the specified curve. [1,2] Sample Find the length of the specified curve. You are given the x values but you need c and d!! Sample Find the length of the specified curve. y = sin(x) [0, ¶ ] Sample Find the length of the specified curve. y = (x2 – 4)2 [0, 4 ] Example A Vertical Tangent 1 3 Find the length of the curve y x between (-1,-1) and (1,1). dy 1 is not defined at x = 0. There is a vertical tangent dx 3x 2 3 at (0,0). Change to x as a function of y in order to make the tangent at the origin horizontal and the derivative equal to zero instead of undefined. dx Solve y x for x. x y and 3y dy 1 3 L 1 1 1 3 y 3 2 2 dy 3.096 using NINT. 2 Sample y = x1/5 [-1,4] Closure Explain the difference in these two formulas. Independent Assignment Notebook: p 485 # 3 - #11 odd. Check your answers in the back of the book. Graded Assignment: HW Sec 7.4 in Schoology. Enter the first ½ of the answers in Schoology. Due Tuesday, May 15, before class.