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)
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 P4P4 P5P5
The length of all segments
Definition of Arc Length
Example Applying the Definition
Example Find the length of the specified curve. =
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 [1,2] Find the length of the specified curve.
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 = (x 2 – 4) 2 [0, 4 ]
Example A Vertical Tangent
Sample y = x 1/5 [-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.