2 Arc Length (#integration, #composition) We have seen that integrals can be used to compute the area between a curve and the 7-axis, and also the volume of a solid of revolution. In this problem, we will explore how to use integrals to compute the arc length of a function. (a) Consider a linear function, f(x) = mx + b in an interval [a, a + Az]. What is the length of the line segment we obtain when graphing f(x) in this interval? How does the expression you found here relate to the derivative of f(x)? (b) What if our function is not a straight line? For example, consider the function g(x) = ³/2 in the interval [0, 2]. Use at least two line segments of your choice to estimate the arc length of g(x) in this interval. (c) Provide a visualization of your method from part (b). (d) We now need an algorithmic way to create various line segments and obtain better estimates. Let us divide the interval [0, 2] into n regions and create a line segment whose value and slope matches that of the left endpoint of g(r) in each region.¹ What would be the equation of l; (r), the line segment in the i-th region in terms of values of a and its derivative, d'?

2 Arc Length (#integration, #composition) We have seen that integrals can be used to compute the area between a curve and the 7-axis, and also the volume of a solid of revolution. In this problem, we will explore how to use integrals to compute the arc length of a function. (a) Consider a linear function, f(x) = mx + b in an interval [a, a + Az]. What is the length of the line segment we obtain when graphing f(x) in this interval? How does the expression you found here relate to the derivative of f(x)? (b) What if our function is not a straight line? For example, consider the function g(x) = ³/2 in the interval [0, 2]. Use at least two line segments of your choice to estimate the arc length of g(x) in this interval. (c) Provide a visualization of your method from part (b). (d) We now need an algorithmic way to create various line segments and obtain better estimates. Let us divide the interval [0, 2] into n regions and create a line segment whose value and slope matches that of the left endpoint of g(r) in each region.¹ What would be the equation of l; (r), the line segment in the i-th region in terms of values of a and its derivative, d'?

Functions and Change: A Modeling Approach to College Algebra (MindTap Course List)

6th Edition

ISBN:9781337111348

Author:Bruce Crauder, Benny Evans, Alan Noell

Publisher:Bruce Crauder, Benny Evans, Alan Noell

ChapterA: Appendix

SectionA.2: Geometric Constructions

Problem 10P: A soda can has a volume of 25 cubic inches. Let x denote its radius and h its height, both in...

See similar textbooks