Chapter 6.4, Problem 17ES

Consider the curve segments $y=x^2$ from $x=\frac{1}{2}$ to $x=2$ and $y=\sqrt{x}$ from $x=\frac{1}{4}$ to $x=4$ .

(a) Graph the two curve segments and use your graphs to explain why the lengths of these two curve segments should be equal.

(b) Set up integrals that give the arc lengths of the curve segments by integrating with respect to $x$ . Demonstrate a substitution that verifies that these two integrals are equal.

(c) Set up integrals that give the arc lengths of the curve segments by integrating with respect to $y$ .

(d) Approximate the arc length of each curve segment using Formula (2) with $n=10$ equal subintervals.

(e) Which of the two approximations in part (d) is more accurate? Explain.

(f) Use the midpoint approximation with $n=10$ subintervals to approximate each arc length integral in part (b).

(g) Use a calculating utility with numerical integration capabilities to approximate the arc length integrals in part (b) to four decimal places.