Length of a Curve

A major use of derivatives and integrals is finding the length of a curve. There's a formula for this:

Figure 11.1: Formula to calculate the length of a curve

The preceding formula contains an integral and a derivative. To find the length of a curve, we'll need both our derivative and integral functions. Copy and paste them into your code if you don't have them yet:

from math import sqrt
 
def derivative(f,x):
    """Returns the value of the derivative of     the function at a given x-value."""
    delta_x = 1/1000000
    return (f(x+delta_x) - f(x))/delta_x
 
def trap_integral(f,a,b,num):
    """Returns the sum of num trapezoids     under f between a and b"""
    width = (b-a)/num
    area = 0.5*width*(f(a) + f(b) + 2*sum([f(a+width*n) \      ...