Adding up tiny straight pieces

You will write three small functions that estimate the length of a curve. The program joins points of the curve by straight pieces, adds the lengths, and counts how many pieces give a chosen accuracy. The checker uses a line and the curve $y = \cosh x$.

The idea. One straight piece from $(x_1, y_1)$ to $(x_2, y_2)$ has length $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$. Cut $[a, b]$ into $n$ equal steps and add the lengths of the pieces. More pieces give a better estimate.

Your tasks.

1. Finish piece_length(x1, y1, x2, y2). Return the length of the straight piece.
2. Finish curve_length(f, a, b, n). Join the $n + 1$ points of f at x = a, a + h, ..., b by straight pieces, with h = (b - a) / n, and return the total length.
3. Finish pieces_needed(f, a, b, exact, tol). Try n = 1, 2, 4, 8, ... and return the first n whose estimate is within tol of exact.