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Calculus Mrs. Cartledge Arc length Sec 7.4 Phantom’s Revenge Kennywood Park

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Objectives Determine the length of a curve.

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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.

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Sum of Disks as the Volume (revisited)

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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.

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The length of one segment P4P4 P5P5

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The length of all segments

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Definition of Arc Length

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Example Applying the Definition

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Example Find the length of the specified curve. =

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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.

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Sample [1,2] Find the length of the specified curve.

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Sample Find the length of the specified curve. You are given the x values but you need c and d!!

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Sample Find the length of the specified curve. y = sin(x) [0, ¶ ]

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Sample Find the length of the specified curve. y = (x 2 – 4) 2 [0, 4 ]

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Example A Vertical Tangent

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Sample y = x 1/5 [-1,4]

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Closure Explain the difference in these two formulas.

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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.

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