Chapter 6.1, Problem 71E

Another look at the Fundamental Theorem

71. Use Exercise 69 to prove that if two trails start at the same place and finish at the same place, then regardless of the ups and downs of the trails, they have the same net change in elevation.

69. Suppose that f and g have continuous derivatives on an interval [a, b]. Prove that if f(a) = g(a) and f(b) = g(b), then ${\displaystyle {\int }_a^b{f}^{\prime }(x)dx={\displaystyle {\int }_a^b{g}^{\prime }(x)dx}}$.