#### Overview of this book

Are you looking to start developing artificial intelligence applications? Do you need a refresher on key mathematical concepts? Full of engaging practical exercises, The Statistics and Calculus with Python Workshop will show you how to apply your understanding of advanced mathematics in the context of Python. The book begins by giving you a high-level overview of the libraries you'll use while performing statistics with Python. As you progress, you'll perform various mathematical tasks using the Python programming language, such as solving algebraic functions with Python starting with basic functions, and then working through transformations and solving equations. Later chapters in the book will cover statistics and calculus concepts and how to use them to solve problems and gain useful insights. Finally, you'll study differential equations with an emphasis on numerical methods and learn about algorithms that directly calculate values of functions. By the end of this book, you’ll have learned how to apply essential statistics and calculus concepts to develop robust Python applications that solve business challenges.
Preface
1. Fundamentals of Python
Free Chapter
2. Python's Main Tools for Statistics
3. Python's Statistical Toolbox
4. Functions and Algebra with Python
5. More Mathematics with Python
6. Matrices and Markov Chains with Python
7. Doing Basic Statistics with Python
8. Foundational Probability Concepts and Their Applications
9. Intermediate Statistics with Python
10. Foundational Calculus with Python
11. More Calculus with Python
12. Intermediate Calculus with Python

# Length of a Spiral

What about spirals, which are expressed in polar coordinates, where r, the distance from the origin, is a function of the theta (θ) angle that's made with the x axis? We can't use our x and y functions to measure the spiral shown in the following diagram:

Figure 11.10: An Archimedean spiral

What we have in the preceding diagram is a spiral that starts at (5,0) and makes 7.5 turns, ending at (11,π). The formula for that curve is r(θ) = 5 + 0.12892θ. The number of radians turned is 7.5 times 2π, which is 15π. We're going to use the same idea as in the previous section: we're going to find the length of the straight line from r(θ) to r(θ+step) for some tiny step in the central angle, as shown in the following diagram:

Figure 11.11: Approximating the length of a tiny part of the curve

The opposite side to the central angle of the triangle shown in the...