Arc Length (#integration, #composition )
Question 2: Arc Length (#integration, #composition ) We have seen that integrals can be used to compute the area between a curve and the x-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 + ∆x]. 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) = 1 3 x3/2 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(x) in each region.1 What would be the equation of ℓi(x), the line segment in the i-th region in terms of values of g and its derivative, g′? (e) Using your work from part (a), calculate the length of each line segment ℓi(x). What would be an expression for the total length in the interval [0, 2]? (f) Use the concepts of #limitscontinutiy to devise an expression for the exact arc length. Then, convert that expression into an integral. (g) Finally, use the integral to compute the arc length of g(x) = 1 3 x3/2 in the interval from [0, 2]. (h) (Optional) Derive an approach to compute the surface area of the solid of revolution obtained by rotating the function g(x) = 1 3 x3/2 in the interval from [0, 2] around the x-axis. Make sure to clearly explain your steps.
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